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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,7)(3,11,6,9)(4,10,5,14)(12,16,13,15)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32)(34,35)(36,43)(40,42)(44,45);(1,3,8,6)(2,9,7,11)(4,12,5,13)(10,15,14,16)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25)(33,34,38,35)(36,41,43,39)(37,40,46,42)(44,48,45,47);(1,4,8,5)(2,10,7,14)(3,12,6,13)(9,15,11,16)(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32)(33,35,38,34)(36,41,43,39)(37,42,46,40)(44,48,45,47);(1,5,8,4)(2,10,7,14)(3,13,6,12)(9,15,11,16)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32)(33,36,38,43)(34,39,35,41)(37,45,46,44)(40,48,42,47);(1,6,8,3)(2,11,7,9)(4,13,5,12)(10,16,14,15)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30);(1,7,8,2)(3,11,6,9)(4,14,5,10)(12,16,13,15)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)(33,37,38,46)(34,40,35,42)(36,44,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gi,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, 8343947);