#6501366

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499814
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

1735 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)(3,6)(4,5)(7,8)(17,18)(19,28)(20,32)(21,26)(22,27)(23,37)(24,30)(25,35)(29,44)(31,40)(33,41)(34,42)(36,39)(38,47)(43,46)(45,48);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,24,27,28)(20,29,34,45)(21,36,23,31)(25,38,41,43)(26,40,37,39)(32,48,42,44)(33,47,35,46);(9,11)(10,13)(12,16)(14,15)(17,20,22,34)(18,25,27,41)(19,29,30,45)(21,35,23,33)(24,38,28,43)(26,42,37,32)(31,47,36,46)(39,48,40,44);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(19,30)(24,28)(29,45)(31,36)(38,43)(39,40)(44,48)(46,47);(10,14)(13,15)(17,21,22,23)(18,26,27,37)(19,31,30,36)(20,33,34,35)(24,39,28,40)(25,32,41,42)(29,46,45,47)(38,44,43,48)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6408562);