#6581265

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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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)(17,18,21,24)(19,23,29,39)(20,25,34,40)(22,26,35,41)(27,38,30,36)(28,37,31,42)(32,44,33,43)(45,48,46,47);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,21,29)(18,23,24,39)(20,31,34,28)(22,27,35,30)(25,42,40,37)(26,38,41,36)(32,45,33,46)(43,47,44,48);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(18,24)(19,27)(20,32)(22,35)(23,36)(25,43)(28,31)(29,30)(33,34)(38,39)(40,44)(47,48);(1,4,5,3)(2,6,8,7)(17,20)(18,25)(19,28)(21,34)(22,33)(23,37)(24,40)(26,43)(27,45)(29,31)(30,46)(32,35)(36,47)(38,48)(39,42)(41,44);(19,27)(20,33)(22,35)(23,38)(25,43)(26,41)(29,30)(32,34)(36,39)(40,44)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,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, 6488461);