#6700785

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1404 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)(11,14)(12,13)(15,16)(17,18,21,23)(19,27,29,24)(20,31,33,25)(22,26,35,37)(28,38,36,40)(30,39,45,43)(32,41,34,42)(44,48,46,47);(1,3,4,8)(2,5,6,7)(18,23)(19,28)(20,32)(22,35)(24,38)(25,41)(27,40)(29,36)(30,45)(31,42)(33,34)(47,48);(2,6)(5,7)(10,13)(14,16)(17,19,21,29)(18,24,23,27)(20,30,33,45)(22,36,35,28)(25,39,31,43)(26,38,37,40)(32,46,34,44)(41,48,42,47);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,23)(19,29)(25,31)(26,37)(28,36)(30,45)(41,42)(44,46);(3,8)(5,7)(17,20,21,33)(18,25,23,31)(19,30,29,45)(22,32,35,34)(24,39,27,43)(26,42,37,41)(28,44,36,46)(38,47,40,48)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=fg,e^2=i,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, 6607981);