#7184828

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1389 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)(10,14)(12,15)(17,18,22,27)(19,28,30,24)(20,25,35,45)(21,36,39,26)(23,37,41,31)(29,43,33,42)(32,48,47,40)(34,46,38,44);(1,3,4,7)(2,5,6,8)(9,10,13,14)(11,12,16,15)(17,19,22,30)(18,23,27,41)(20,32,35,47)(21,37,39,31)(24,36,28,26)(25,42,45,43)(29,38,33,34)(40,46,48,44);(2,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,24)(20,33)(21,31)(22,30)(23,26)(25,43)(27,28)(29,35)(32,38)(34,47)(36,41)(37,39)(40,44)(42,45)(46,48);(10,14)(11,16)(17,20,22,35)(18,25,27,45)(19,29,30,33)(21,38,39,34)(23,40,41,48)(24,42,28,43)(26,46,36,44)(31,47,37,32);(9,12,13,15)(10,11,14,16)(19,30)(23,41)(24,28)(29,33)(31,37)(32,47)(40,48)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092024);