#6367559

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367534
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

1583 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)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18,23,29)(19,25,32,41)(20,26,35,44)(21,28,22,27)(24,36,39,31)(30,40,47,42)(33,48,46,38)(34,45,37,43);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19)(18,24)(20,33)(21,31)(22,36)(23,32)(25,27)(26,42)(28,41)(29,39)(30,34)(35,46)(37,47)(38,43)(40,44)(45,48);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,19)(18,25)(20,30)(21,36)(22,31)(23,32)(24,27)(26,40)(28,39)(29,41)(33,34)(35,47)(37,46)(38,43)(42,44)(45,48);(17,20,23,35)(18,26,29,44)(19,30,32,47)(21,34,22,37)(24,38,39,48)(25,40,41,42)(27,43,28,45)(31,46,36,33);(2,8)(3,10)(7,9)(11,16)(17,21,23,22)(18,27,29,28)(19,31,32,36)(20,34,35,37)(24,25,39,41)(26,43,44,45)(30,46,47,33)(38,40,48,42)

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ei,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, 6274755);