Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,8)(9,10,12,11)(13,14)(15,20)(16,26)(17,27)(18,22)(19,24)(21,23)(25,28);(1,2,3,4)(6,8)(9,11,12,10)(13,15,23,16,18,25,17,24)(14,19,27,28,22,26,21,20);(2,4)(5,6,7,8)(13,15,17,24,18,25,23,16)(14,19,21,20,22,26,27,28);(13,16,23,25,18,24,17,15)(14,20,27,19,22,28,21,26)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=f,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274755);