#6730878

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 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)(20,32)(21,25)(22,26)(23,31)(24,33)(28,38)(29,37)(30,36)(34,45)(35,40)(39,46)(41,44)(42,47)(43,48);(1,3,4,8)(2,5,6,7)(17,19)(18,23)(20,29)(21,31)(22,28)(24,37)(25,27)(26,36)(30,35)(32,41)(33,44)(34,42)(38,40)(39,47)(43,46)(45,48);(10,14)(15,16)(19,28)(20,33)(22,35)(23,36)(24,32)(26,40)(27,38)(29,43)(30,31)(37,48)(41,47)(42,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,21,33)(18,24,25,32)(19,29,31,44)(22,34,35,46)(23,37,27,41)(26,39,40,45)(28,42,30,43)(36,47,38,48);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,25)(19,30)(22,35)(23,36)(24,32)(27,38)(28,31)(29,43)(34,46)(37,47)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=gi,e^2=g,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, 6638074);