#9159919

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=hi,e^2=g,f^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, 9067115);