#6356837

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356733
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

1535 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)(3,5)(4,11)(6,9)(7,13)(8,10)(12,16)(14,15)(17,18)(19,28)(20,25)(21,26)(22,37)(23,27)(24,31)(29,44)(30,39)(32,45)(33,48)(34,41)(35,43)(36,42)(38,46)(40,47);(1,3,10,9,6,8,5,2)(4,11,13,16,14,15,12,7)(17,19,22,31,23,32,21,30)(18,24,26,28,27,39,37,45)(20,33,34,47,35,29,36,46)(25,40,42,48,43,38,41,44);(1,4,6,14)(2,7,9,16)(3,11,8,15)(5,12,10,13);(1,5,6,10)(2,3,9,8)(4,12,14,13)(7,11,16,15)(17,20,23,35)(18,25,27,43)(19,29,32,33)(21,36,22,34)(24,38,39,40)(26,42,37,41)(28,44,45,48)(30,47,31,46);(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,35,36)(24,28,39,45)(25,41,43,42)(29,46,33,47)(38,48,40,44)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283373
maximal normal subgroup of	#72283373

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264033);