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 | |
6085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,4)(3,6,7,8)(10,13)(14,16)(17,18,20,22)(19,21,23,24);(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,23)(22,24);(2,4)(6,8)(10,13)(14,16)(18,22)(19,23);(3,7)(6,8)(10,13)(11,15)(19,23)(21,24);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=h,e^2=ghi,f^2=hi,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.
| maximal subgroup of | #72256162, #72262948, #72262959, … |
| maximal normal subgroup of | #72256162, #72262948, #72262959, … |
| maximal quotient of | #72266259 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7998946);