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 | |
4068 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(34,37)(36,44)(39,42)(41,47);(2,7)(3,10)(8,13)(11,16)(18,19)(23,25)(24,26)(30,31)(33,34,38,37)(35,39,43,42)(36,40,44,45)(41,46,47,48);(2,7)(4,9)(8,13)(12,15)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35,38,43)(34,39,37,42)(36,41,44,47)(40,46,45,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,15)(14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36)(34,40)(35,41)(37,45)(38,44)(39,46)(42,48)(43,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,34)(35,42,43,39)(36,45,44,40)(41,48,47,46)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026043);