#7090493

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1652 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,21)(19,27,32,30)(20,25,34,33)(22,37,28,26)(24,38,36,31)(29,41,44,43)(35,47,42,40)(39,48,46,45);(9,10)(11,15)(12,14)(13,16)(17,19,28,38,23,32,22,31)(18,24,37,30,21,36,26,27)(20,29,42,48,34,44,35,45)(25,39,47,43,33,46,40,41);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,34)(18,25,21,33)(19,29,32,44)(22,35,28,42)(24,39,36,46)(26,40,37,47)(27,41,30,43)(31,45,38,48);(11,16)(13,15)(17,21,23,18)(19,30,32,27)(20,33,34,25)(22,26,28,37)(24,31,36,38)(29,43,44,41)(35,40,42,47)(39,45,46,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(20,34)(25,33)(29,44)(35,42)(39,46)(40,47)(41,43)(45,48)

Polycyclic presentation:

2^9|a^2=i,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=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997689);