#6366752

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366710
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

1570 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273948);