#9232788

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=hi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9139984);