#6566269

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535696
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61256	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1604 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,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, 6473465);