#6819551

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1283 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,29,35)(20,24,32,38)(21,25,28,34)(27,33,39,43)(30,36,42,46)(31,37,41,45)(40,44,47,48);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(19,27)(20,31)(21,28)(23,33)(24,37)(25,34)(29,39)(30,42)(32,41)(35,43)(36,46)(38,45);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,19,28,39,22,29,21,27)(18,23,34,43,26,35,25,33)(20,30,41,47,32,42,31,40)(24,36,45,48,38,46,37,44);(10,14)(13,15)(19,29)(23,35)(27,39)(30,42)(33,43)(36,46)(40,47)(44,48);(17,20,21,31,22,32,28,41)(18,24,25,37,26,38,34,45)(19,30,27,40,29,42,39,47)(23,36,33,44,35,46,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6726747);