Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,21)(23,24);(1,2,3,4)(9,11)(10,14)(12,19)(13,18)(15,21)(16,22)(17,23)(20,24);(10,15)(12,20)(14,21)(16,17)(19,24)(22,23);(11,18)(14,22)(19,24)(21,23);(10,16)(14,22)(15,17)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=di,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72249651, #72250828, #72250855, … |
| maximal normal subgroup of | #72249651, #72250828, #72250855, … |
| maximal quotient of | #72264386, #72315994, #72316245 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277992);