#6818259

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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

1497 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)(17,18,23,26)(19,27,25,21)(20,24,31,22)(28,30,32,29);(2,5)(3,7)(17,19)(18,21)(20,22)(23,25)(24,31)(26,27)(28,29)(30,32);(9,11,12,16)(10,13,14,15)(17,20,21,29,23,31,27,30)(18,24,19,28,26,22,25,32);(2,5)(6,8)(17,21,23,27)(18,25,26,19)(20,30,31,29)(22,32,24,28);(1,3,4,7)(2,6,5,8)(17,22,21,32,23,24,27,28)(18,20,19,29,26,31,25,30)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6725455);