#6358455

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358426
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1681 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)(3,6)(4,7)(5,9)(8,12)(10,13)(11,15)(14,16)(17,18,23,28)(19,29,32,43)(20,25,35,42)(21,27,22,26)(24,36,39,31)(30,45,47,40)(33,48,46,38)(34,44,37,41);(1,3,5,11)(2,6,9,15)(4,10,12,16)(7,13,8,14)(17,19,23,32)(18,24,28,39)(20,33,35,46)(21,31,22,36)(25,40,42,45)(26,29,27,43)(30,37,47,34)(38,44,48,41);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15);(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,41)(27,44)(28,42)(29,45)(31,46)(32,47)(33,36)(39,48)(40,43);(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,35,37)(24,29,39,43)(25,41,42,44)(30,46,47,33)(38,45,48,40)

Polycyclic presentation:

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

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, 6265651);