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 | |
6410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,20,22)(19,21,23,24);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,15,16,13)(17,18)(19,21)(20,22)(23,24);(2,5)(6,8)(10,16)(11,12);(9,11,14,12)(10,15,16,13);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,13,16,15)(17,19,20,23)(18,21,22,24);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,16)(14,15)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72234851, #72234852, #72234895, … |
| maximal normal subgroup of | #72234851, #72234852, #72234895, … |
| maximal quotient of | #72260843, #72316104, #72316474 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7577692);