#7090338

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	#30398	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1665 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)(19,29)(20,34)(21,26)(22,38)(23,28)(24,33)(25,31)(27,40)(30,46)(32,44)(35,42)(36,48)(37,41)(39,45)(43,47);(9,10)(11,15)(12,14)(13,16)(17,19,22,33,23,32,40,47)(18,24,27,29,28,43,38,44)(20,30,35,39,31,21,36,37)(25,41,42,26,34,45,48,46);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,31)(18,25,28,34)(19,30,32,21)(22,35,40,36)(24,41,43,45)(26,44,46,29)(27,42,38,48)(33,39,47,37);(11,16)(13,15)(17,21,23,30)(18,26,28,46)(19,31,32,20)(22,39,40,37)(24,42,43,48)(25,44,34,29)(27,41,38,45)(33,35,47,36);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,32)(20,31)(24,43)(25,34)(29,44)(33,47)(35,36)(42,48)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,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,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, 6997534);