#6357731

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357716
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1837 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^c=eh,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264927);