#6436835

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433077
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=h,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,g^c=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344031);