#6544041

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2748 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:

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

Polycyclic presentation:

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

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, 6451237);