#6489615

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396811);