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 | |
5783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,4)(3,6,7,8)(9,10,13,12)(11,14,15,16)(17,18,21,24)(19,22,20,23);(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24);(2,4)(6,8)(10,12)(11,15);(3,7)(6,8)(10,12)(14,16);(1,3,5,7)(2,6,4,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,20)(18,22,24,23);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,16,15,14)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^9|a^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72232554, #72256306, #72263022, … |
| maximal normal subgroup of | #72232554, #72256306, #72263022, … |
| maximal quotient of | #72232589, #72336621, #72336975 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7974380);