Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476082 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,8)(9,10)(11,12)(13,14,17,21,18,16,19,22)(15,23,25,28,26,24,27,20);(1,2)(3,4)(10,11)(13,15,18,26)(14,20,16,28)(17,27,19,25)(21,24,22,23);(2,4)(9,11,12,10)(13,16,19,21,18,14,17,22)(15,24,27,28,26,23,25,20);(6,8)(10,11)(13,14,18,16)(15,23,26,24)(17,22,19,21)(20,27,28,25)Polycyclic presentation:
2^9|a^2=efi,b^2=i,c^2=ei,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383716);