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 | |
4146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18)(19,24)(20,28)(21,26)(22,29)(23,25)(27,31)(30,32)(33,34)(35,41)(36,42)(37,43)(38,44)(39,40)(45,46)(47,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19)(18,24)(20,25)(21,22)(23,28)(26,29)(27,32)(30,31)(34,40)(37,44)(41,42)(45,47);(2,6)(7,14)(8,11)(15,16)(18,25)(19,28)(22,31)(26,32)(33,35,39,36)(34,41,40,42)(37,45,44,47)(38,48,43,46);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,20,23,24)(18,19,25,28)(21,29,30,27)(22,26,31,32)(33,36,39,35)(34,42,40,41)(37,45,44,47)(38,48,43,46);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21)(18,26)(19,22)(20,27)(23,30)(24,29)(25,32)(28,31)(33,37,39,44)(34,43,40,38)(35,45,36,47)(41,46,42,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,22)(18,27)(19,21)(20,26)(23,31)(24,32)(25,29)(28,30)(33,38,39,43)(34,44,40,37)(35,46,36,48)(41,45,42,47)Polycyclic presentation:
2^9|c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8832396);