Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,9)(10,11)(13,14,19,15)(16,24,26,28)(17,22,18,21)(20,27,23,25);(1,2,3,4)(5,6,8,9)(7,11,12,10)(13,15,19,14)(16,23,26,20)(17,22,18,21)(24,25,28,27);(2,4)(13,16)(14,20)(15,23)(17,27)(18,25)(19,26)(21,28)(22,24);(5,7,8,12)(6,10,9,11)(13,16)(14,20)(15,23)(17,25)(18,27)(19,26)(21,24)(22,28)Polycyclic presentation:
2^9|a^2=i,b^2=egi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 345188);