#6350962

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2736 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)(3,7,10,15)(4,6,12,9)(11,13,16,14)(17,18,24,29)(19,30,20,34)(21,36,37,26)(22,28,23,27)(25,33,41,32)(31,46,35,45)(38,43,39,44)(40,47,42,48);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,24,20)(18,25,29,41)(21,31,37,35)(22,33,23,32)(26,40,36,42)(27,34,28,30)(38,48,39,47)(43,46,44,45);(1,3,5,10)(2,7,8,15)(4,11,12,16)(6,13,9,14)(17,20,24,19)(18,25,29,41)(21,31,37,35)(22,32,23,33)(26,42,36,40)(27,34,28,30)(38,48,39,47)(43,45,44,46);(2,8)(3,10)(6,9)(11,16)(17,21)(18,26)(19,31)(20,35)(22,38)(23,39)(24,37)(25,40)(27,43)(28,44)(29,36)(30,45)(32,47)(33,48)(34,46)(41,42);(21,37)(26,36)(31,35)(38,39)(40,42)(43,44)(45,46)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258158);