#6587417

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1346 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,25)(19,27,30,23)(20,24,34,41)(22,26,35,42)(28,38,31,36)(29,43,45,37)(32,39,33,40)(44,48,46,47);(1,3,5,4)(2,6,7,8)(17,19,21,30)(18,23,25,27)(20,29,34,45)(22,28,35,31)(24,37,41,43)(26,36,42,38)(32,44,33,46)(39,47,40,48);(10,13)(14,16)(19,28)(20,32)(22,35)(23,36)(24,39)(26,42)(27,38)(30,31)(33,34)(40,41)(44,46)(47,48);(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,29)(21,34)(22,32)(23,37)(25,41)(26,39)(27,43)(28,44)(30,45)(31,46)(33,35)(36,47)(38,48)(40,42);(1,4,5,3)(2,6,7,8)(19,28)(20,33)(22,35)(23,36)(24,40)(26,42)(27,38)(29,45)(30,31)(32,34)(37,43)(39,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=ehi,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, 6494613);