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 | |
5792 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)(3,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,21,23,24);(1,3)(2,4)(5,7)(6,8)(17,18)(19,21)(20,22)(23,24);(1,4,6,5)(2,3,7,8)(10,13)(14,16)(17,19,20,23)(18,21,22,24);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16);(1,5,6,4)(2,3,7,8)(9,11,12,15)(10,14,13,16);(1,3)(2,5)(4,7)(6,8)Polycyclic presentation:
2^9|a^2=gi,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72317637, #72318169, #72319613, … |
| maximal normal subgroup of | #72317637, #72318169, #72319613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7603243);