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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,23,26)(19,27,24,20)(21,25,31,22)(28,32,30,29)(33,34,39,43)(35,40,38,42)(36,46,37,41)(44,48,45,47);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,15)(14,16)(17,19,23,24)(18,20,26,27)(21,30,31,28)(22,29,25,32)(33,35,39,38)(34,40,43,42)(36,45,37,44)(41,48,46,47);(2,6)(4,13)(7,9)(10,14)(21,31)(22,25)(28,30)(29,32)(33,36,39,37)(34,41,43,46)(35,44,38,45)(40,47,42,48);(1,3)(2,7)(4,14)(5,11)(6,9)(8,16)(10,13)(12,15)(17,20,23,27)(18,24,26,19)(21,32,31,29)(22,30,25,28)(33,37,39,36)(34,41,43,46)(35,45,38,44)(40,47,42,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,21)(18,25)(19,28)(20,29)(22,26)(23,31)(24,30)(27,32)(33,38,39,35)(34,42,43,40)(36,44,37,45)(41,47,46,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,13)(14,16)(17,22,23,25)(18,21,26,31)(19,29,24,32)(20,30,27,28)(33,34)(35,40)(36,46)(37,41)(38,42)(39,43)(44,48)(45,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948042);