#6366701

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366542
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

1533 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273897);