#6356654

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356633
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1577 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:

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

Polycyclic presentation:

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

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, 6263850);