Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,7)(10,11)(13,14,17,22)(15,19,18,21)(16,24,25,20)(23,28,26,27);(1,2)(3,4)(5,7,8,6)(9,10)(11,12)(13,15,17,18)(14,19,22,21)(16,23,25,26)(20,27,24,28);(2,4)(6,7)(9,10,12,11)(13,16,17,25)(14,20,22,24)(15,23,18,26)(19,27,21,28);(9,11,12,10)(14,21)(15,18)(19,22)(20,28)(23,26)(24,27)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=ei,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329823);