#7083452

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1341 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,30)(19,31,35,43)(20,26,32,46)(21,27,39,48)(22,29,23,28)(25,33,44,34)(36,42,37,45)(38,41,40,47);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,33,24,35,23,34)(18,25,28,43,30,44,29,31)(20,21,37,40,32,39,36,38)(26,41,42,48,46,47,45,27);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,24,32)(18,26,30,46)(19,21,35,39)(22,37,23,36)(25,41,44,47)(27,43,48,31)(28,42,29,45)(33,40,34,38);(1,4,5,3)(2,7,8,6)(11,16)(13,15)(17,21)(18,27)(19,32)(20,35)(22,38)(23,40)(24,39)(25,42)(26,43)(28,47)(29,41)(30,48)(31,46)(33,37)(34,36)(44,45);(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,36,32,37)(21,38,39,40)(25,43,44,31)(26,45,46,42)(27,47,48,41)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 6990648);