#6359224

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359112
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

1516 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)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(17,18,24,29)(19,30,34,46)(20,26,37,21)(22,28,23,27)(25,41,44,33)(31,47,35,32)(36,39,40,42)(38,48,45,43);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19)(18,25)(20,35)(21,38)(22,33)(23,41)(24,34)(26,45)(27,30)(28,46)(29,44)(31,37)(32,39)(36,43)(40,48)(42,47);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,20)(18,26)(19,31)(21,29)(22,40)(23,36)(24,37)(25,38)(27,39)(28,42)(30,47)(32,46)(33,43)(34,35)(41,48)(44,45);(17,21,24,26)(18,20,29,37)(19,32,34,47)(22,39,23,42)(25,43,44,48)(27,36,28,40)(30,31,46,35)(33,45,41,38);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,22,24,23)(18,27,29,28)(19,33,34,41)(20,36,37,40)(21,39,26,42)(25,30,44,46)(31,48,35,43)(32,45,47,38)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266420);