#6368376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368046
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

1497 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)(3,6)(4,12)(5,9)(7,11)(8,13)(10,15)(14,16)(17,18)(19,27)(20,24)(21,34)(22,25)(23,30)(26,37)(28,45)(29,43)(31,48)(32,44)(33,41)(35,39)(36,42)(38,47)(40,46);(1,3,6,2)(4,10,14,8)(5,9,7,11)(12,13,16,15)(17,19,22,29)(18,23,25,39)(20,31,33,47)(21,35,37,30)(24,40,41,45)(26,43,34,27)(28,36,46,32)(38,44,48,42);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,38)(29,39)(31,40)(32,42)(33,41)(34,37)(35,43)(36,44)(45,47)(46,48);(1,5,6,7)(2,9,3,11)(4,13,14,15)(8,16,10,12)(17,20,22,33)(18,24,25,41)(19,28,29,46)(21,36,37,32)(23,38,39,48)(26,44,34,42)(27,45,43,40)(30,47,35,31);(2,3)(8,10)(9,11)(12,16)(18,25)(19,29)(24,41)(26,34)(28,46)(30,35)(31,47)(42,44)

Polycyclic presentation:

2^9|b^2=hi,d^2=hi,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=ehi,f^b=fh,e^c=eh,g^c=gi,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, 6275572);