#9159906

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

4034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=hi,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9067102);