#7091213

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1251 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,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, 6998409);