#6434386

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1377 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)(19,29)(20,34)(21,26)(22,27)(23,28)(24,33)(25,35)(30,38)(31,40)(32,39)(36,42)(37,41)(43,45)(44,48)(46,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,33,23,31,21,32)(18,24,26,29,28,39,27,40)(20,30,36,46,37,44,35,45)(25,38,41,43,42,48,34,47);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,22,36,23,37,21,35)(18,25,26,41,28,42,27,34)(19,30,33,46,31,44,32,45)(24,38,29,43,39,48,40,47);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(19,31)(24,39)(29,40)(30,44)(32,33)(38,48)(43,47)(45,46);(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,29,39,40)(25,41,42,34)(30,45,44,46)(38,43,48,47)

Polycyclic presentation:

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

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, 6341582);