#6369137

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369070
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

1556 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)(3,7)(4,10)(5,9)(6,11)(8,12)(13,16)(14,15)(17,18,24,29)(19,30,34,46)(20,21,26,39)(22,28,23,27)(25,41,45,33)(31,32,35,48)(36,42,40,38)(37,47,43,44);(1,3)(2,7)(4,14)(5,12)(6,13)(8,9)(10,15)(11,16)(17,19,24,34)(18,25,29,45)(20,35,26,31)(21,37,39,43)(22,33,23,41)(27,30,28,46)(32,42,48,38)(36,47,40,44);(1,3)(2,8)(4,12)(5,14)(6,13)(7,9)(10,16)(11,15)(17,20,24,26)(18,21,29,39)(19,31,34,35)(22,40,23,36)(25,43,45,37)(27,42,28,38)(30,32,46,48)(33,47,41,44);(1,3)(2,7)(4,14)(5,12)(6,13)(8,9)(10,15)(11,16)(17,21)(18,26)(19,32)(20,29)(22,38)(23,42)(24,39)(25,44)(27,40)(28,36)(30,35)(31,46)(33,43)(34,48)(37,41)(45,47);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,15,16,8)(17,22,24,23)(18,27,29,28)(19,33,34,41)(20,36,26,40)(21,38,39,42)(25,30,45,46)(31,44,35,47)(32,43,48,37)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276333);