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 | #2671 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,10)(9,12)(11,16)(17,18,23,28)(19,24,30,21)(20,26,32,25)(22,31,29,27)(34,39)(35,42)(37,45)(43,47);(1,2)(3,8)(4,12)(5,10)(6,7)(9,14)(11,15)(13,16)(17,19,23,30)(18,24,28,21)(20,29,32,22)(25,31,26,27)(33,34,38,39)(35,40,42,46)(36,37,41,45)(43,44,47,48);(17,20,23,32)(18,25,28,26)(19,22,30,29)(21,31,24,27)(33,35,38,42)(34,40,39,46)(36,43,41,47)(37,44,45,48);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,13,12,15)(17,20)(18,26)(19,29)(21,31)(22,30)(23,32)(24,27)(25,28)(33,35)(34,40)(36,43)(37,44)(38,42)(39,46)(41,47)(45,48);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,15,8,13)(17,21)(18,19)(20,31)(22,25)(23,24)(26,29)(27,32)(28,30)(33,36)(34,37)(35,43)(38,41)(39,45)(40,44)(42,47)(46,48);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,11,12,16)(17,22,23,29)(18,27,28,31)(19,20,30,32)(21,26,24,25)(33,37)(34,41)(35,44)(36,39)(38,45)(40,47)(42,48)(43,46)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=ghi,e^2=ghi,f^2=gh,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, 9770601);