#6830186

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1411 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)(11,14)(12,13)(15,16)(17,18,22,26)(19,23,29,36)(20,24,33,40)(21,25,28,35)(27,34,41,45)(30,37,32,39)(31,38,43,47)(42,46,44,48);(1,3,4,8)(2,5,6,7)(19,27)(20,31)(21,28)(23,34)(24,38)(25,35)(29,41)(30,32)(33,43)(36,45)(37,39)(40,47);(10,13)(14,16)(17,19,28,41,22,29,21,27)(18,23,35,45,26,36,25,34)(20,32,43,42,33,30,31,44)(24,39,47,46,40,37,38,48);(2,6)(5,7)(19,29)(23,36)(27,41)(30,32)(34,45)(37,39)(42,44)(46,48);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,20,28,43,22,33,21,31)(18,24,35,47,26,40,25,38)(19,30,41,44,29,32,27,42)(23,37,45,48,36,39,34,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,e^2=ghi,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, 6737382);