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 | |
4002 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)(8,15)(10,13)(11,16)(17,18,23,25)(19,24,28,30)(20,29,27,22)(21,26,31,32)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(1,2,6,7)(3,8,9,15)(4,12,14,5)(10,11,13,16)(17,19)(18,24)(20,26)(21,22)(23,28)(25,30)(27,32)(29,31)(34,40)(35,44)(37,47)(42,45);(2,7)(5,12)(8,15)(11,16)(18,25)(19,28)(22,29)(26,32)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,3)(2,8)(4,13)(5,11)(6,9)(7,15)(10,14)(12,16)(17,20,23,27)(18,22,25,29)(19,26,28,32)(21,30,31,24)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,4)(2,5)(3,10)(6,14)(7,12)(8,16)(9,13)(11,15)(17,21)(18,26)(19,22)(20,24)(23,31)(25,32)(27,30)(28,29)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,5)(2,4)(3,11)(6,12)(7,14)(8,13)(9,16)(10,15)(17,22)(18,27)(19,21)(20,25)(23,29)(24,32)(26,30)(28,31)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=gh,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, 8833521);