#6434004

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433044
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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1576 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,30,39)(20,31,34,45)(21,26,22,36)(25,32,41,33)(28,42,44,47)(29,38,35,40)(37,46,48,43);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,30)(18,24,27,39)(20,28,34,44)(21,35,22,29)(25,37,41,48)(26,40,36,38)(31,42,45,47)(32,46,33,43);(11,15)(13,16)(17,20,21,32,23,34,22,33)(18,25,36,45,27,41,26,31)(19,28,35,46,30,44,29,43)(24,37,38,47,39,48,40,42);(10,14)(13,16)(17,21,23,22)(18,26,27,36)(19,29,30,35)(20,32,34,33)(24,38,39,40)(25,31,41,45)(28,43,44,46)(37,47,48,42);(1,3,4,7)(2,6,5,8)(19,30)(24,39)(28,44)(29,35)(37,48)(38,40)(42,47)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,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, 6341200);