Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
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 | |
22374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8);(5,6)(7,8)(10,14)(15,21)(16,22)(23,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24);(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72249625, #72256860, #72257736, … |
| maximal normal subgroup of | #72249625, #72256860, #72257736, … |
| maximal quotient of | #72264082, #72264621, #72315727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7539394);