#72319214

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂³×#102
prime radical	#41×#135	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=f,d^2=hj,f^2=h,g^2=i,i^2=j,c^a=cj,c^b=cg,d^b=dj,d^c=dh,e^b=eh,e^d=ehj,f^c=fj,g^b=gi,g^c=gij,i^b=ij,i^c=ij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319265, #72319510, #72319749, …
maximal normal subgroup of	#72319265, #72319510, #72319749, …

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, 2, F.6);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);