#6579633

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1432 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,27)(22,28)(23,26)(29,38)(30,40)(31,41)(32,39)(33,42)(34,45)(35,43)(36,44)(37,46)(47,48);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,33)(21,29)(22,37)(23,32)(25,42)(26,39)(27,38)(28,46)(30,36)(31,35)(34,47)(40,44)(41,43)(45,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20)(18,25)(19,29)(21,35)(22,34)(23,36)(24,38)(26,44)(27,43)(28,45)(30,31)(32,47)(33,37)(39,48)(40,41)(42,46);(18,26)(20,34)(21,36)(22,35)(24,39)(25,27)(29,30)(31,37)(33,47)(38,42)(40,48)(44,45);(17,21,23,34)(18,27,26,45)(19,30,32,33)(20,35,36,22)(24,40,39,42)(25,43,44,28)(29,31,47,37)(38,41,48,46)

Polycyclic presentation:

2^9|b^2=fg,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,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, 6486829);