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 | |
4079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,25)(19,24,27,22)(20,21,26,32)(28,29,31,30)(33,34,40,44)(35,45,48,41)(36,37,38,42)(39,47,46,43);(4,11)(8,14)(10,15)(12,16)(17,19)(18,24)(20,28)(21,29)(22,25)(23,27)(26,31)(30,32)(33,35)(34,41)(36,43)(37,46)(38,47)(39,42)(40,48)(44,45);(2,7)(3,9)(8,14)(10,15)(18,25)(19,27)(20,26)(29,30)(33,36,40,38)(34,42,44,37)(35,43,48,47)(39,45,46,41);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,26)(18,21,25,32)(19,28,27,31)(22,30,24,29)(33,37,40,42)(34,36,44,38)(35,46,48,39)(41,43,45,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,26)(19,29)(20,25)(22,28)(23,32)(24,31)(27,30)(33,38,40,36)(34,42,44,37)(35,47,48,43)(39,45,46,41);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,27)(19,25)(20,30)(21,31)(23,24)(26,29)(28,32)(33,39,40,46)(34,43,44,47)(35,42,48,37)(36,45,38,41)Polycyclic presentation:
2^9|a^2=gi,c^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9690301);