#6364913

order	512
nilpotency class	3
isoclinism	isoclinic class of #6364762
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

1505 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^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=ei,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, 6272109);