#6767451

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 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)(19,28)(20,32)(21,27)(22,26)(23,38)(24,30)(25,36)(29,45)(31,41)(33,48)(34,43)(35,44)(37,40)(39,47)(42,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,33)(21,37)(22,30)(23,31)(25,42)(26,28)(27,41)(29,35)(32,45)(34,46)(36,47)(38,40)(39,43)(44,48);(10,13)(14,16)(17,20,22,35)(18,25,26,43)(19,29,30,33)(21,36,23,34)(24,39,28,42)(27,44,38,32)(31,47,37,46)(40,48,41,45);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,26)(19,30)(25,43)(27,38)(29,33)(31,37)(32,44)(46,47);(3,8)(5,7)(10,13)(14,16)(17,21,22,23)(18,27,26,38)(19,31,30,37)(20,34,35,36)(24,40,28,41)(25,32,43,44)(29,46,33,47)(39,45,42,48)

Polycyclic presentation:

2^9|a^2=fi,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=efhi,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, 6674647);