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 | |
4124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,10)(5,12,13,8)(11,16,15,14)(17,18,22,26)(19,25,21,23)(20,24,28,31)(27,32,29,30)(33,34)(35,37)(36,44)(38,40)(39,42)(41,47)(43,48)(45,46);(3,9)(7,10)(11,15)(14,16)(17,19,22,21)(18,23,26,25)(20,27,28,29)(24,30,31,32)(33,35,38,42)(34,39,40,37)(36,45,47,43)(41,46,44,48);(2,4)(7,10)(8,12)(14,16)(20,28)(24,31)(27,29)(30,32)(34,40)(35,42)(41,44)(43,45);(1,3,6,9)(2,7,4,10)(5,11,13,15)(8,14,12,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36,38,47)(34,41,40,44)(35,43,42,45)(37,46,39,48);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,34)(35,39)(36,41)(37,42)(38,40)(43,48)(44,47)(45,46);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,21,22,19)(18,25,26,23)(20,29,28,27)(24,32,31,30)(33,37,38,39)(34,42,40,35)(36,46,47,48)(41,45,44,43)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8001739);