#7083378

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

1343 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)(17,18,24,29)(19,30,34,48)(20,35,31,26)(21,27,39,47)(22,28,23,40)(25,32,44,33)(36,43,37,45)(38,46,41,42);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,32,24,34,23,33)(18,25,40,48,29,44,28,30)(20,21,37,41,31,39,36,38)(26,42,45,27,35,46,43,47);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,24,31)(18,26,29,35)(19,21,34,39)(22,37,23,36)(25,42,44,46)(27,30,47,48)(28,43,40,45)(32,41,33,38);(1,4,5,3)(2,7,8,6)(11,16)(13,15)(17,21)(18,27)(19,31)(20,34)(22,38)(23,41)(24,39)(25,43)(26,30)(28,46)(29,47)(32,37)(33,36)(35,48)(40,42)(44,45);(10,14)(13,15)(17,22,24,23)(18,28,29,40)(19,32,34,33)(20,36,31,37)(21,38,39,41)(25,30,44,48)(26,45,35,43)(27,46,47,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gh,d^2=gi,e^2=i,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,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, 6990574);