#6970576

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(18,20)(19,25)(22,26)(23,30)(27,31)(28,32);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,18,21,20)(19,23,28,27)(22,24,26,29)(25,31,32,30);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19,26,32,21,28,22,25)(18,23,29,30,20,27,24,31);(2,6)(5,7)(10,13)(14,16)(17,20,21,18)(19,27,28,23)(22,29,26,24)(25,30,32,31);(9,11,12,15)(10,14,13,16)(18,20)(23,27)(24,29)(30,31)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285261
maximal normal subgroup of	#72285261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6877772);