#72287754

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23001
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#33675	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8293 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,11)(4,5)(6,9)(7,10)(8,12)(13,16)(14,15)(17,18,23,28)(19,24,21,26)(20,31,34,48)(22,36,38,40)(25,41,43,32)(27,35,45,30)(29,46,33,42)(37,39,47,44);(1,3,6,13,7,14,5,12)(2,8,4,15,10,16,9,11)(17,19,23,21)(18,24,28,26)(20,29,34,33)(22,30,38,35)(25,39,43,44)(27,40,45,36)(31,46,48,42)(32,37,41,47);(1,4,6,2,7,9,5,10)(3,8,13,11,14,16,12,15)(17,20,21,33,23,34,19,29)(18,25,24,39,28,43,26,44)(22,37,35,32,38,47,30,41)(27,42,40,31,45,46,36,48);(1,5,7,6)(3,12,14,13)(18,26)(19,21)(20,32,34,41)(22,35)(24,28)(25,42,43,46)(29,47,33,37)(30,38)(31,39,48,44)(36,40)

Polycyclic presentation:

2^10|a^2=j,b^2=gi,c^2=e,d^2=egh,e^2=i,f^2=hj,g^2=ij,h^2=j,b^a=bg,c^a=cf,c^b=cgij,d^a=dghj,d^b=dj,d^c=dfhij,e^a=ej,e^d=ej,f^a=fh,f^c=fhj,f^d=fhi,g^a=gij,g^c=gij,h^a=hj,h^c=hj,h^d=hj

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287776
maximal normal subgroup of	#72287776

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 2, F.7*F.9);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.5*F.7*F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);