#7019259

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1735 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)(3,5,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,28,21,27)(19,25,34,36,30,39,31,40)(20,26,35,37,33,41,32,42)(29,38,45,46,44,48,43,47);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(18,24)(20,32)(21,23)(25,36)(27,28)(29,43)(31,34)(33,35)(37,42)(39,40)(44,45)(46,47);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,25,28,39)(20,29,33,44)(21,31,23,34)(24,36,27,40)(26,38,41,48)(32,43,35,45)(37,46,42,47);(11,16)(13,15)(17,20,21,32,22,33,23,35)(18,26,27,42,28,41,24,37)(19,29,31,43,30,44,34,45)(25,38,40,47,39,48,36,46);(10,14)(13,15)(19,30)(20,33)(25,39)(26,41)(31,34)(32,35)(36,40)(37,42)

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=i,d^2=f,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6926455);