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 | |
4820 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)(15,22)(20,23)(21,24);(2,4)(9,10)(11,20)(12,15)(13,17)(14,24)(16,22)(18,21)(19,23);(6,8)(9,11)(10,15)(12,21)(13,22)(14,23)(16,18)(17,19)(20,24);(9,12,13,23)(10,16,17,20)(11,21,22,14)(15,18,19,24);(10,17)(15,19)(16,20)(18,24)Polycyclic presentation:
2^9|a^2=f,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, #72249641, #72249643 |
| maximal normal subgroup of | #72249548, #72249641, #72249643 |
| maximal quotient of | #72264572, #72307883, #72310623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344890);