#6572565

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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	#62958	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2746 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=h,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,f^b=fi,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, 6479761);