#7008395

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#45023	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1574 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)(17,18,23,25,21,26,22,27)(19,28,30,32,20,24,29,31);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,21)(22,29)(23,30)(25,31)(26,28)(27,32);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(18,25)(22,23)(24,31)(26,27)(28,32)(29,30);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,20,21,19)(18,24,26,28)(22,30,23,29)(25,31,27,32);(10,14)(13,15)(19,20)(24,28)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=gi,c^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6915591);