#6369200

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369150
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

1544 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:

(3,10)(4,5)(7,15)(8,13)(11,16)(12,14)(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);(1,2)(3,11)(4,13)(5,8)(6,9)(7,12)(10,16)(14,15)(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)(2,7)(4,10)(5,14)(6,12)(8,16)(9,11)(13,15)(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);(1,2)(3,7)(4,13)(5,8)(6,9)(10,15)(11,12)(14,16)(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)(2,8,9,13)(3,10,12,14)(7,16,11,15)(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=h,b^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,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, 6276396);