#6484247

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1462 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=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, 6391443);