#7092141

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1353 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)(19,29)(20,25)(21,26)(22,37)(23,28)(24,34)(27,30)(31,44)(32,42)(33,46)(35,40)(36,45)(38,39)(41,47)(43,48);(9,10)(11,15)(12,14)(13,16)(17,19,30,47,23,33,22,34)(18,24,37,46,28,41,27,29)(20,31,48,36,32,21,35,38)(25,39,40,26,42,45,43,44);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,32)(18,25,28,42)(19,31,33,21)(22,35,30,48)(24,39,41,45)(26,29,44,46)(27,43,37,40)(34,38,47,36);(11,16)(13,15)(17,21,23,31)(18,26,28,44)(19,32,33,20)(22,38,30,36)(24,40,41,43)(25,29,42,46)(27,45,37,39)(34,35,47,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,33)(20,32)(24,41)(25,42)(29,46)(34,47)(35,48)(40,43)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,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=dfi,e^b=ehi,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, 6999337);