#6714577

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667508
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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1487 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:

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

Polycyclic presentation:

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

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, 6621773);