Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417692 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6,9,8)(7,11,12,10)(13,14,19,24,17,22,18,23)(15,25,26,28,21,16,20,27);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,21)(18,20)(19,26)(22,25)(23,28)(24,27);(2,4)(5,8,9,6)(7,11,12,10)(13,16)(14,20)(15,24)(17,25)(18,28)(19,27)(21,23)(22,26);(2,4)(7,12)(10,11)(13,16)(14,21)(15,22)(17,25)(18,27)(19,28)(20,23)(24,26)Polycyclic presentation:
2^9|a^2=efhi,c^2=efi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357408);