#7181312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,d^2=fi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7088508);