#7090369

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	#30414	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1655 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)(20,25)(21,26)(22,37)(23,27)(24,31)(28,39)(30,45)(32,46)(33,47)(34,41)(35,44)(36,40)(38,43)(42,48);(9,10)(11,15)(12,14)(13,16)(17,19,22,31,23,32,39,48)(18,24,28,29,27,42,37,46)(20,33,34,38,30,21,35,36)(25,43,44,47,45,40,41,26);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,30)(18,25,27,45)(19,21,32,33)(22,34,39,35)(24,40,42,43)(26,46,47,29)(28,44,37,41)(31,36,48,38);(11,16)(13,15)(17,21,23,33)(18,26,27,47)(19,30,32,20)(22,38,39,36)(24,41,42,44)(25,29,45,46)(28,40,37,43)(31,34,48,35);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,27)(20,30)(21,33)(24,42)(28,37)(29,46)(34,35)(36,38)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,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=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, 6997565);