#7027392

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#31036	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=fg,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,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, 6934588);