#7026480

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	#24248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1708 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,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=ehi,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, 6933676);