#6573281

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480477);