#6820327

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806575
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

1393 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)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,28)(21,23)(22,26)(25,30)(27,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(18,23)(19,20)(22,29)(24,25)(27,30)(28,31);(9,12,13,11)(10,14,15,16)(17,19,22,20,21,27,29,30)(18,24,26,28,23,31,32,25);(1,3,4,7)(2,6,5,8)(18,23)(24,31)(25,28)(26,32);(2,5)(6,8)(17,20,29,19,21,30,22,27)(18,25,32,31,23,28,26,24)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6727523);