#6581293

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1529 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,30,37)(20,31,22,25)(21,26,33,42)(28,36,34,39)(29,38,45,47)(32,41,35,40)(43,46,44,48);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(19,28)(20,22)(21,32)(24,36)(25,31)(26,41)(30,34)(33,35)(37,39)(40,42)(43,44)(46,48);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,19,23,30)(18,24,27,37)(20,28,22,34)(21,29,33,45)(25,39,31,36)(26,38,42,47)(32,43,35,44)(40,48,41,46);(1,4,5,3)(2,6,8,7)(17,20)(18,25)(21,33)(22,23)(24,37)(27,31)(28,34)(29,44)(38,46)(40,41)(43,45)(47,48);(17,21,23,33)(18,26,27,42)(19,29,30,45)(20,32,22,35)(24,38,37,47)(25,40,31,41)(28,43,34,44)(36,46,39,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6488489);