#7083442

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1361 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)(19,31,35,44)(20,36,32,26)(21,27,40,46)(22,29,23,28)(25,33,45,34)(37,43,38,47)(39,48,41,42);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,33,24,35,23,34)(18,25,28,44,30,45,29,31)(20,21,38,41,32,40,37,39)(26,42,47,27,36,48,43,46);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20,24,32)(18,26,30,36)(19,21,35,40)(22,38,23,37)(25,42,45,48)(27,31,46,44)(28,47,29,43)(33,41,34,39);(1,4,5,3)(2,7,8,6)(11,16)(13,15)(17,21)(18,27)(19,32)(20,35)(22,39)(23,41)(24,40)(25,43)(26,31)(28,42)(29,48)(30,46)(33,38)(34,37)(36,44)(45,47);(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,37,32,38)(21,39,40,41)(25,44,45,31)(26,43,36,47)(27,42,46,48)

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=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, 6990638);