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 | |
4092 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)(3,12)(4,5)(6,16)(7,9)(8,13)(10,15)(11,14)(17,18)(19,26)(20,24)(21,25)(22,27)(23,30)(28,32)(29,31)(33,34)(35,43)(36,40)(37,47)(38,42)(39,44)(41,45)(46,48);(1,3,7,13)(2,8,9,12)(4,14,15,6)(5,16,10,11)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,32)(26,27)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,47,48,41);(2,9)(3,13)(4,15)(11,16)(17,20,22,21)(18,24,27,25)(19,28,30,29)(23,31,26,32)(33,34,38,42)(35,39,44,43)(36,40,46,48)(37,41,45,47);(1,4)(2,10)(3,14)(5,9)(6,13)(7,15)(8,11)(12,16)(17,21,22,20)(18,25,27,24)(19,29,30,28)(23,32,26,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,44)(45,47)(46,48);(1,5,7,10)(2,4,9,15)(3,11,13,16)(6,12,14,8)(17,19)(18,26)(20,28)(21,29)(22,30)(23,27)(24,32)(25,31)(33,36,38,46)(34,40,42,48)(35,37,44,45)(39,41,43,47);(1,6,7,14)(2,11,9,16)(3,4,13,15)(5,12,10,8)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,37)(34,41)(35,36)(38,45)(39,40)(42,47)(43,48)(44,46)Polycyclic presentation:
2^9|b^2=h,c^2=gi,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9632024);