Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,23)(22,24);(2,4)(9,10)(11,16)(12,19)(13,17)(14,23)(15,22)(18,20)(21,24);(6,8)(9,11,13,21)(10,15,17,19)(12,20,22,14)(16,18,24,23);(9,12)(10,16)(11,20)(13,22)(14,21)(15,18)(17,24)(19,23);(10,17)(15,19)(16,24)(18,23)Polycyclic presentation:
2^9|a^2=f,c^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249548, #72249635, #72249645, … |
| maximal normal subgroup of | #72249548, #72249635, #72249645, … |
| maximal quotient of | #72264787, #72310679, #72310688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344888);