#72322152

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623254	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17336 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2227 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322260, #72322814
maximal normal subgroup of	#72322260, #72322814

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