#6364625

order	512
nilpotency class	3
isoclinism	isoclinic class of #6364250
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

1496 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^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,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271821);