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 |
| prime radical | #35250 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4772 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)(11,15)(12,16)(13,18)(14,17)(19,20)(21,22)(23,24);(2,4)(9,10)(11,19)(12,21)(13,23)(14,17)(15,20)(16,22)(18,24);(6,8)(9,11,14,12)(10,15,17,16)(13,22,24,20)(18,21,23,19);(9,12,14,11)(10,16,17,15)(13,20,24,22)(18,19,23,21);(10,17)(15,16)(18,23)(19,21)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^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, #72249625, #72249630 |
| maximal normal subgroup of | #72249548, #72249625, #72249630 |
| maximal quotient of | #72264590, #72310676, #72310695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344892);