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 | #48690 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,12,10,6)(4,15,14,8)(5,9,13,16)(17,18,23,27)(19,24,29,31)(20,21,25,30)(22,26,28,32)(33,34,40,44)(35,39,37,42)(36,46,45,41)(38,43,47,48);(1,3,7,10)(2,6,11,12)(4,13,14,5)(8,16,15,9)(17,19)(18,24)(20,22)(21,26)(23,29)(25,28)(27,31)(30,32)(33,35,40,37)(34,39,44,42)(36,47,45,38)(41,43,46,48);(1,4,7,14)(2,8,11,15)(3,13,10,5)(6,16,12,9)(17,20,23,25)(18,21,27,30)(19,28,29,22)(24,32,31,26)(33,36,40,45)(34,41,44,46)(35,38,37,47)(39,48,42,43);(1,4)(2,8)(3,13)(5,10)(6,16)(7,14)(9,12)(11,15)(17,20)(18,21)(19,22)(23,25)(24,26)(27,30)(28,29)(31,32)(33,37,40,35)(34,42,44,39)(36,38,45,47)(41,48,46,43);(1,5,7,13)(2,9,11,16)(3,14,10,4)(6,15,12,8)(17,21)(18,25)(19,26)(20,27)(22,31)(23,30)(24,28)(29,32)(33,38)(34,43)(35,45)(36,37)(39,41)(40,47)(42,46)(44,48);(1,6)(2,10)(3,11)(4,16)(5,8)(7,12)(9,14)(13,15)(17,22)(18,26)(19,20)(21,24)(23,28)(25,29)(27,32)(30,31)(33,39,40,42)(34,35,44,37)(36,48,45,43)(38,46,47,41)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9048867);