#7091782

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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1357 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)(18,24)(21,22)(25,39)(26,40)(27,37)(28,42)(30,38)(32,44)(34,35)(41,45)(43,48)(46,47);(9,10)(11,15)(12,14)(13,16)(17,18,22,24,23,27,21,37)(19,28,30,42,31,25,38,39)(20,32,35,44,36,40,34,26)(29,45,46,41,33,48,47,43);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,33)(21,38)(22,30)(23,31)(24,39)(26,43)(27,28)(29,36)(32,45)(34,46)(35,47)(37,42)(40,48)(41,44);(11,16)(13,15)(17,20,23,36)(18,26,27,44)(19,29,31,33)(21,35,22,34)(24,40,37,32)(25,41,28,43)(30,47,38,46)(39,45,42,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,24,27,37)(19,30,31,38)(20,34,36,35)(25,42,28,39)(26,32,44,40)(29,46,33,47)(41,45,43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ghi,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=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, 6998978);