Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454426 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,11,12,10)(14,20)(16,26)(17,18)(21,25)(22,28)(23,24);(1,2)(3,4)(6,9)(10,11)(13,14,17,20,19,24,18,23)(15,25,26,21,27,28,16,22);(2,4)(5,7,8,12)(6,10,9,11)(13,15)(14,21)(16,17)(18,26)(19,27)(20,25)(22,24)(23,28);(7,12)(10,11)(13,16,18,27,19,26,17,15)(14,22,23,28,24,21,20,25)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=efi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367313);