Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1420 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)(15,19)(16,21)(17,25)(18,22)(20,23)(24,28)(26,27);(5,7)(6,10)(8,11)(9,12)(13,15,18,24)(14,19,22,28)(16,23,17,26)(20,25,27,21);(5,8,9,6)(7,11,12,10)(13,15)(14,20)(16,23)(17,26)(18,24)(19,25)(21,28)(22,27);(2,4)(7,12)(10,11)(13,16,18,17)(14,21,22,25)(15,23,24,26)(19,20,28,27)Polycyclic presentation:
2^9|a^2=eh,b^2=i,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356825);