#6529045

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
prime radical	#62831	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,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, 6436241);