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 |
| prime radical | #50530 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,20)(19,21,24,26)(22,28,31,25)(27,30,32,29)(33,34,38,41)(35,42,37,39)(36,44,46,40)(43,47,45,48);(4,11)(8,14)(10,15)(12,16)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,35,38,37)(34,39,41,42)(36,43,46,45)(40,47,44,48);(2,7)(3,9)(8,14)(10,15)(18,20)(19,24)(25,28)(27,32)(33,35,38,37)(34,39,41,42)(36,45,46,43)(40,48,44,47);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,30,32,29)(33,36,38,46)(34,40,41,44)(35,43,37,45)(39,47,42,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,26)(18,24,20,19)(22,30,31,29)(25,27,28,32)(33,37,38,35)(34,39,41,42)(36,45,46,43)(40,47,44,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9746514);