Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1228 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)(13,14,18,23)(15,25,16,20)(17,24,19,22)(21,26,28,27);(5,7)(6,10)(8,11)(9,12)(13,15,19,26,18,16,17,27)(14,20,22,21,23,25,24,28);(5,8,9,6)(7,11,12,10)(13,16)(14,21)(15,18)(17,27)(19,26)(20,22)(23,28)(24,25);(2,4)(7,12)(10,11)(13,17,18,19)(14,22,23,24)(15,26,16,27)(20,28,25,21)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=ei,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 345091);