#6356591

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356533
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

1583 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263787);