#7093470

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

1366 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)(19,31,21,27)(20,26,37,44)(22,29,23,28)(25,33,40,34)(32,42,35,45)(36,46,38,43)(39,48,41,47);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,33,24,21,23,34)(18,25,28,27,30,40,29,31)(20,35,38,47,37,32,36,48)(26,41,46,42,44,39,43,45);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,32)(21,35)(22,38)(23,36)(24,37)(25,39)(27,45)(28,46)(29,43)(30,44)(31,42)(33,48)(34,47)(40,41);(11,16)(13,15)(17,21,24,19)(18,27,30,31)(20,32,37,35)(22,33,23,34)(25,29,40,28)(26,42,44,45)(36,48,38,47)(39,46,41,43);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,21,34)(20,36,37,38)(25,27,40,31)(26,43,44,46)(32,47,35,48)(39,42,41,45)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=ghi,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=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, 7000666);