#6506801

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500434
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6413997);