#6820356

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	C₂×#493	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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)(19,27)(24,32)(28,30)(29,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,18)(19,28)(20,25)(21,23)(22,26)(24,31)(27,30)(29,32);(9,12,13,11)(10,14,15,16)(17,19,23,31,22,27,20,29)(18,24,25,28,26,32,21,30);(1,3,4,7)(2,6,5,8)(17,20,22,23)(18,25,26,21)(19,29,27,31)(24,28,32,30);(2,5)(6,8)(17,21,20,26,22,25,23,18)(19,30,29,32,27,28,31,24)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=fi,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, 6727552);