Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,21)(17,20)(18,22)(23,27)(24,28)(25,26);(1,2,3,4)(15,23,24,25)(19,26,28,27);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,25)(17,23)(18,24)(20,27)(21,26)(22,28);(2,4)(14,20)(16,17)(19,27)(21,22)(23,25)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,g^d=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332005);