#6729849

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1454 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6637045);