#9159917

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

4209 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18,22,23)(19,24,27,30)(20,25,31,28)(21,29,32,26)(33,34)(35,44)(36,38)(37,42)(39,43)(40,47)(41,46)(45,48);(2,7)(4,8)(9,13)(12,14)(17,18,22,23)(19,24,27,30)(20,28,31,25)(21,26,32,29)(33,35)(34,40)(36,45)(37,46)(38,42)(39,47)(41,48)(43,44);(2,7)(3,10)(9,13)(11,16)(18,23)(19,27)(25,28)(26,29)(33,36,39,46)(34,38,43,41)(35,45,47,37)(40,42,44,48);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31)(33,36)(34,41)(35,45)(37,47)(38,43)(39,46)(40,48)(42,44);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,20)(18,25)(19,26)(21,24)(22,31)(23,28)(27,29)(30,32)(33,37,39,45)(34,42,43,48)(35,46,47,36)(38,44,41,40);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21)(18,26)(19,28)(20,30)(22,32)(23,29)(24,31)(25,27)(33,38,39,41)(34,36,43,46)(35,42,47,48)(37,40,45,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 9067113);