#6357567

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357552
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

1860 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:

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

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,d^b=di,e^d=eh

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, 6264763);