#6767473

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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)(17,18,22,28)(19,27,23,24)(20,31,34,42)(21,35,37,26)(25,33,40,29)(30,38,36,41)(32,45,47,44)(39,43,48,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,28,27)(20,29,34,33)(21,36,37,30)(25,31,40,42)(26,41,35,38)(32,46,47,43)(39,45,48,44);(10,13)(14,16)(17,20,22,34)(18,25,28,40)(19,29,23,33)(21,32,37,47)(24,31,27,42)(26,39,35,48)(30,43,36,46)(38,44,41,45);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,30)(20,32)(22,37)(23,36)(24,38)(25,39)(27,41)(28,35)(29,43)(31,44)(33,46)(34,47)(40,48)(42,45);(3,8)(5,7)(10,13)(14,16)(17,19,22,23)(18,27,28,24)(20,33,34,29)(21,36,37,30)(25,31,40,42)(26,38,35,41)(32,43,47,46)(39,45,48,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,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=efhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6674669);