#6364121

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363930
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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=i,e^2=h,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,e^b=ei,f^b=fh,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, 6271317);