#6529591

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
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×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2731 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)(19,27,30,40)(20,25)(21,23)(22,26)(24,29,35,34)(28,41,42,48)(31,36)(32,39)(33,37)(38,43,46,45)(44,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(18,23)(22,33)(24,35)(25,36)(29,34)(32,44)(38,46)(43,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,28)(21,30)(22,34)(23,35)(25,38)(26,40)(27,37)(29,33)(31,42)(32,45)(36,46)(39,48)(41,47)(43,44);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,25)(19,28)(21,31)(22,32)(23,36)(24,38)(26,39)(27,41)(29,43)(30,42)(33,44)(34,45)(35,46)(37,47)(40,48);(2,8)(6,9)(7,16)(14,15)(20,31)(25,36)(28,42)(32,44)(38,46)(39,47)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=h,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,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, 6436787);