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 | #2714 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3939 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)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(34,39)(35,42)(36,44)(46,47);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(18,19)(23,25)(24,26)(30,31)(33,34,38,39)(35,40,42,37)(36,45,44,41)(43,47,48,46);(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35)(34,40)(36,43)(37,39)(38,42)(41,46)(44,48)(45,47);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,38,44)(34,41,39,45)(35,43,42,48)(37,47,40,46);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,40)(34,35,39,42)(36,46,44,47)(41,48,45,43)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8343987);