#6460635

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1635 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367831);