Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3633 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)(9,10,12,16,13,17,20,15)(11,14,19,23,18,22,24,21);(5,6)(7,8)(9,10)(11,14)(12,15)(13,17)(16,20)(18,22)(19,21)(23,24);(2,4)(6,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24);(2,4)(10,15)(12,20)(14,21)(16,17)(19,24)(22,23);(11,18)(14,22)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,f^b=fi,e^c=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72279716, #72280210, #72308183, … |
| maximal normal subgroup of | #72279716, #72280210, #72308183, … |
| maximal quotient of | #72312948, #72313131, #72313136 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408193);