#6844096

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#26968	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,22,24)(19,21,27,26)(20,23,30,31)(25,29,32,28);(9,11)(10,13)(12,16)(14,15)(17,19,22,27)(18,21,24,26)(20,28,30,29)(23,25,31,32);(2,6)(7,8)(11,16)(13,15)(17,20,18,23,22,30,24,31)(19,25,26,28,27,32,21,29);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,18,22,24)(19,21,27,26)(20,23,30,31)(25,29,32,28);(1,4,5,3)(2,8,6,7)(17,21,18,19,22,26,24,27)(20,29,23,25,30,28,31,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fh,d^2=gi,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6751292);