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 | |
4117 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(19,25)(22,26)(27,31)(30,32)(33,34,39,44)(35,40,47,46)(36,43,38,41)(37,42,45,48);(2,5)(7,9)(8,12)(14,15)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(33,35)(34,40)(36,37)(38,45)(39,47)(41,48)(42,43)(44,46);(1,3,6,11)(2,7,5,9)(4,10,13,16)(8,14,12,15)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,36,39,38)(34,41,44,43)(35,45,47,37)(40,42,46,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,37,39,45)(34,42,44,48)(35,36,47,38)(40,43,46,41);(1,4,6,13)(2,8,5,12)(3,10,11,16)(7,14,9,15)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,38,39,36)(34,43,44,41)(35,45,47,37)(40,42,46,48);(1,5,6,2)(3,9,11,7)(4,12,13,8)(10,15,16,14)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,34)(35,46)(36,41)(37,42)(38,43)(39,44)(40,47)(45,48)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664191);