#6969825

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#25397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1424 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)(3,6)(4,8)(5,7)(10,14)(11,12)(17,18,22,26,23,27,31,32)(19,28,21,30,29,24,25,20);(1,2,6,3)(4,7,5,8)(9,10,13,14)(11,16,12,15)(17,19)(18,24)(20,26)(21,22)(23,29)(25,31)(27,28)(30,32);(1,3,6,2)(4,7,5,8)(9,11,13,12)(10,15,14,16)(17,20,23,30)(18,25,27,21)(19,26,29,32)(22,24,31,28);(1,4,6,5)(2,7,3,8)(9,12,13,11)(10,15,14,16)(17,21,23,25)(18,20,27,30)(19,22,29,31)(24,26,28,32);(1,5,6,4)(2,7,3,8)(17,22,23,31)(18,26,27,32)(19,25,29,21)(20,24,30,28)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6877021);