Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 | #35222 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
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)(19,22)(20,23)(21,24);(2,4)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24);(6,8)(9,11,14,12)(10,15,17,16)(13,20,21,19)(18,23,24,22);(9,12,14,11)(10,16,17,15)(13,19,21,20)(18,22,24,23);(10,17)(15,16)(18,24)(22,23)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,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72306123, #72306136, #72306145, … |
| maximal normal subgroup of | #72306123, #72306136, #72306145, … |
| maximal quotient of | #72306420, #72306496, #72308706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344864);