Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
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 | |
1152 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)(6,10)(11,12)(13,14)(15,16)(17,22)(18,23)(19,20)(21,27)(24,28)(25,26);(2,4)(5,6,9,10)(7,11,8,12)(13,15,18,25,17,20,28,21)(14,19,23,27,22,16,24,26);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,22)(17,19)(18,27)(21,24)(23,25)(26,28);(5,8,9,7)(6,11,10,12)(13,16)(14,21)(15,23)(17,19)(18,26)(20,24)(22,25)(27,28)Polycyclic presentation:
2^9|b^2=efg,c^2=ef,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^c=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346494);