#6529058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499849
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

2782 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6436254);