#6529048

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499844
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2812 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)(20,33)(21,27)(22,28)(23,26)(24,31)(25,36)(30,39)(32,41)(34,42)(35,43)(37,40)(38,44)(45,47)(46,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,24)(20,34)(21,37)(22,31)(23,32)(25,42)(26,40)(27,29)(28,41)(30,35)(33,45)(36,46)(38,47)(39,44)(43,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20,23,35)(18,25,26,44)(19,30,32,34)(21,36,22,38)(24,39,40,42)(27,43,28,33)(29,45,41,48)(31,46,37,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,26)(20,35)(24,40)(27,28)(29,41)(30,34)(36,38)(46,47);(2,8)(6,9)(7,16)(14,15)(17,21,23,22)(18,27,26,28)(19,31,32,37)(20,36,35,38)(24,41,40,29)(25,43,44,33)(30,46,34,47)(39,48,42,45)

Polycyclic presentation:

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

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, 6436244);