#6818190

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1515 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(19,28)(21,29)(24,31)(26,30);(2,5)(3,7)(17,18)(19,29)(20,25)(21,28)(22,27)(23,32)(24,26)(30,31);(9,11,12,16)(10,13,14,15)(17,19,23,26,22,28,20,30)(18,24,25,29,27,31,32,21);(2,5)(6,8)(17,20,22,23)(18,25,27,32)(19,26,28,30)(21,31,29,24);(1,3,4,7)(2,6,5,8)(17,21,23,24,22,29,20,31)(18,26,25,28,27,30,32,19)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6725386);