#6572620

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

2750 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6479816);