#6745577

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2302 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)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,21,23)(19,26,27,32)(20,25,22,30)(24,28,31,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(18,23)(19,27)(25,30)(28,29);(11,15)(13,16)(17,19,21,27)(18,24,23,31)(20,29,22,28)(25,32,30,26);(2,6)(5,7)(17,18)(19,24)(20,30)(21,23)(22,25)(26,29)(27,31)(28,32);(3,8)(5,7)(11,15)(13,16)(17,20,21,22)(18,25,23,30)(19,28,27,29)(24,26,31,32)

Polycyclic presentation:

2^9|a^2=h,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,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, 6652773);