#6820348

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1380 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)(19,28)(20,27)(21,25)(22,26)(23,31)(24,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,18,22,26)(19,21,31,32)(20,24,29,27)(23,25,28,30);(9,12,13,11)(10,14,15,16)(17,19,29,25,22,31,20,30)(18,23,24,32,26,28,27,21);(1,3,4,7)(2,6,5,8)(17,20,22,29)(18,24,26,27)(19,30,31,25)(21,23,32,28);(2,5)(6,8)(17,21,29,23,22,32,20,28)(18,25,24,31,26,30,27,19)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6727544);