Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4600 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)(9,10)(11,13)(12,15)(14,16)(17,18)(19,23)(20,21)(22,24);(6,8)(10,11)(12,16)(18,21)(22,23);(5,6)(7,8)(10,11)(14,15)(19,24)(22,23);(1,2,3,4)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,24)(21,23);(9,12,13,16)(10,14,11,15)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72278026, C2×#1515137, #72300334, … |
| maximal normal subgroup of | #72278026, C2×#1515137, #72300334, … |
| maximal quotient of | #72278119, #72278620, C2×#1515137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1422333);