Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #158815 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
2528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(10,11)(18,19)(22,23);(2,4)(9,10)(11,12)(14,16)(17,18,21,19)(20,23,24,22);(10,11)(13,14,15,16)(17,19,21,18)(20,22,24,23);(1,2)(3,4)(5,6)(7,8)(9,11,12,10)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|a^2=efh,b^2=f,c^2=eg,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de92 conjugacy classes — view the full character table.
| maximal subgroup of | #72299079, #72299085 |
| maximal normal subgroup of | #72299079, #72299085 |
| maximal quotient of | #72299905 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 66546);