#7089949

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1428 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)(17,18,22,26)(19,24,29,42)(20,25,34,44)(21,35,38,27)(23,36,40,30)(28,41,32,43)(31,39,47,48)(33,46,37,45);(9,10)(11,15)(12,14)(13,16)(17,19,22,29)(18,23,26,40)(20,31,34,47)(21,36,38,30)(24,27,42,35)(25,41,44,43)(28,33,32,37)(39,45,48,46);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,32)(21,30)(22,29)(23,35)(25,43)(26,42)(27,40)(28,34)(31,33)(36,38)(37,47)(39,46)(41,44)(45,48);(11,16)(13,15)(17,20,22,34)(18,25,26,44)(19,28,29,32)(21,37,38,33)(23,39,40,48)(24,41,42,43)(27,46,35,45)(30,31,36,47);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,26)(19,29)(25,44)(27,35)(28,32)(30,36)(31,47)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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,f^b=fi,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, 6997145);