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 | |
5839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,11)(14,15)(19,24)(20,22);(9,10,13,11)(12,14,16,15)(17,18)(19,20)(21,23)(22,24);(2,5)(6,8)(10,11)(14,15)(17,19)(18,22)(20,23)(21,24);(3,7)(6,8)(10,11)(12,16)(17,18,21,23)(19,22,24,20);(1,3)(2,6)(4,7)(5,8)(9,11,13,10)(12,15,16,14)(17,20)(18,19)(21,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,20)(21,23)(22,24)Polycyclic presentation:
2^9|b^2=g,d^2=gi,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7995223);