#6992846

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	#39131	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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)(22,27)(23,28)(25,29)(30,38)(32,44)(33,37)(34,41)(35,42)(36,40)(39,43)(45,48)(46,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19,23,22)(18,24,28,27)(20,29,33,36)(21,34,35,30)(25,37,40,31)(26,41,42,38)(32,45,46,43)(39,44,48,47);(3,8)(5,7)(9,11,14,10)(12,15,13,16)(17,20)(18,25)(19,29)(21,32)(22,36)(23,33)(24,37)(26,39)(27,31)(28,40)(30,43)(34,45)(35,46)(38,47)(41,44)(42,48);(2,6)(5,7)(9,12,14,13)(10,15,11,16)(17,21,23,35)(18,26,28,42)(19,30,22,34)(20,32,33,46)(24,38,27,41)(25,39,40,48)(29,43,36,45)(31,44,37,47);(9,13,14,12)(10,15,11,16)(17,19,23,22)(18,27,28,24)(20,29,33,36)(21,34,35,30)(25,31,40,37)(26,38,42,41)(32,45,46,43)(39,47,48,44)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6900042);