#6730752

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 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)(19,27)(20,24)(21,25)(22,26)(23,30)(28,45)(29,43)(31,48)(32,40)(33,41)(34,42)(35,38)(36,44)(37,46)(39,47);(1,3,4,8)(2,5,6,7)(17,19)(18,23)(20,31)(21,30)(22,35)(24,39)(25,27)(26,43)(28,33)(29,34)(32,47)(36,46)(37,41)(38,42)(40,48)(44,45);(10,14)(15,16)(17,20,21,32)(18,24,25,40)(19,28,30,46)(22,36,34,33)(23,37,27,45)(26,44,42,41)(29,47,35,31)(38,48,43,39);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,25)(20,32)(23,27)(26,42)(28,46)(31,47)(33,36)(38,43);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(19,29)(22,34)(23,38)(26,42)(27,43)(28,47)(30,35)(31,46)(33,36)(37,48)(39,45)(41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6637948);