#6350795

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350717
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

2633 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,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, 6257991);