#6368935

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368846
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

1508 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^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,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276131);