Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,8)(7,11,12,10)(13,14,17,20)(15,25,16,27)(18,24,19,23)(21,28,22,26);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(14,20)(15,16)(23,24)(26,28);(2,4)(5,8,9,6)(7,11,12,10)(13,15,17,16)(14,21,20,22)(18,28,19,26)(23,27,24,25);(7,12)(10,11)(13,16,17,15)(14,22,20,21)(18,28,19,26)(23,27,24,25)Polycyclic presentation:
2^9|a^2=e,b^2=gi,c^2=e,d^2=i,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, 345087);