#6913833

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907849
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
prime radical	#36127	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1706 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6821029);