#6508542

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1470 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:

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

Polycyclic presentation:

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

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, 6415738);