#6857322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18,23,27)(19,24,30,40)(20,31,33,41)(21,26,22,36)(25,35,42,32)(28,43,46,48)(29,38,34,39)(37,44,47,45);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15)(17,19)(18,24)(20,28)(21,34)(22,29)(23,30)(25,37)(26,39)(27,40)(31,43)(32,45)(33,46)(35,44)(36,38)(41,48)(42,47);(9,10,13,12)(11,15,16,14)(17,20,21,35,23,33,22,32)(18,25,36,41,27,42,26,31)(19,28,34,44,30,46,29,45)(24,37,38,48,40,47,39,43);(2,6)(5,7)(17,21,23,22)(18,26,27,36)(19,29,30,34)(20,32,33,35)(24,38,40,39)(25,41,42,31)(28,44,46,45)(37,43,47,48);(3,8)(5,7)(9,12,13,10)(11,14,16,15)

Polycyclic presentation:

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

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, 6764518);