Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,12)(7,10,8,11)(13,14,17,21)(15,23,26,24)(16,22,18,20)(19,25,28,27);(1,2,3,4)(5,7,9,8)(6,10,12,11)(13,14,16,20,17,21,18,22)(15,24,25,28,26,23,27,19);(5,7,9,8)(6,11,12,10)(13,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,25)(26,28);(2,4)(5,8,9,7)(6,11,12,10)(13,15,17,26)(14,19,21,28)(16,27,18,25)(20,23,22,24)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=fi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339445);