#6357666

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357620
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

1811 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=h,e^2=hi,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^d=eh

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, 6264862);