#6859411

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1482 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)(22,36)(23,27)(25,33)(28,45)(29,39)(30,40)(32,48)(34,43)(35,41)(37,44)(38,47)(42,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,29)(18,24,27,39)(20,32,34,28)(21,35,22,30)(25,42,44,38)(26,41,36,40)(31,48,43,45)(33,46,37,47);(10,14)(15,16)(17,20)(18,25)(19,28)(21,33)(22,37)(23,34)(24,38)(26,43)(27,44)(29,32)(30,46)(31,36)(35,47)(39,42)(40,45)(41,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(19,29)(24,39)(28,32)(30,35)(38,42)(40,41)(45,48)(46,47);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,26,27,36)(19,30,29,35)(20,33,34,37)(24,40,39,41)(25,43,44,31)(28,46,32,47)(38,45,42,48)

Polycyclic presentation:

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

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, 6766607);