Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(10,11)(13,14,18,21,19,22,16,17)(15,23,24,28,25,27,26,20);(9,10)(11,12)(13,15,19,25)(14,20,22,28)(16,24,18,26)(17,23,21,27);(1,2,3,4)(9,10,12,11)(13,16,19,18)(14,21,22,17)(15,24,25,26)(20,27,28,23);(6,8)(9,11,12,10)(13,17,16,22,19,21,18,14)(15,20,26,27,25,28,24,23)Polycyclic presentation:
2^9|a^2=eg,b^2=i,c^2=f,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341420);