#9159903

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

4206 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)(3,5)(4,12)(6,9)(7,11)(8,13)(10,14)(15,16)(17,18,22,23)(19,24,27,30)(20,25,31,28)(21,29,32,26)(34,38)(35,41)(40,43)(42,44);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,18,22,23)(19,24,27,30)(20,28,31,25)(21,26,32,29)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(4,13)(8,12)(10,16)(14,15)(18,23)(19,27)(25,28)(26,29)(33,35,37,41)(34,39,38,46)(36,42,45,44)(40,47,43,48);(1,4,6,13)(2,8,9,12)(3,10,11,16)(5,15,7,14)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,5,6,7)(2,3,9,11)(4,14,13,15)(8,16,12,10)(17,20)(18,25)(19,26)(21,24)(22,31)(23,28)(27,29)(30,32)(33,36,37,45)(34,40,38,43)(35,42,41,44)(39,47,46,48);(1,3)(2,5)(4,10)(6,11)(7,9)(8,15)(12,14)(13,16)(17,21)(18,26)(19,28)(20,30)(22,32)(23,29)(24,31)(25,27)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 9067099);