#6769530

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

1291 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=hi,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, 6676726);