Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| 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 | #35252 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(11,17)(12,19)(14,21)(16,23)(18,20)(22,24);(1,2,3,4)(9,10)(11,14)(12,16)(13,15)(17,21)(18,22)(19,23)(20,24);(9,11,13,18)(10,14,15,22)(12,20,19,17)(16,24,23,21);(6,8);(10,15)(14,22)(16,23)(21,24)Polycyclic presentation:
2^9|b^2=f,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^b=ei,g^c=gi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72305327, #72305333, #72305343 |
| maximal normal subgroup of | #72305327, #72305333, #72305343 |
| maximal quotient of | #72306189, #72306199, #72306202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536135);