#6767307

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6674503);