Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(17,18,23,22)(19,26,24,20)(21,29,30,25)(27,28,31,32)(33,34)(35,40)(36,37)(38,43)(39,46)(41,42)(44,48)(45,47);(2,6)(7,9)(8,12)(14,15)(17,19,23,24)(18,20,22,26)(21,27,30,31)(25,28,29,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,48)(42,47);(2,6)(3,10)(8,12)(11,16)(17,20,23,26)(18,24,22,19)(21,28,30,32)(25,31,29,27)(33,35)(34,40)(36,47)(37,45)(38,46)(39,43)(41,48)(42,44);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,36,38,42)(34,41,43,37)(35,44,46,47)(39,48,40,45);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,21,23,30)(18,25,22,29)(19,27,24,31)(20,28,26,32)(33,34)(35,39)(36,41)(37,42)(38,43)(40,46)(44,48)(45,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,18)(19,26,24,20)(21,29,30,25)(27,32,31,28)(33,37,38,41)(34,42,43,36)(35,45,46,48)(39,47,40,44)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034688);