Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1264 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)(7,10)(8,9)(11,12)(13,14,19,22,17,21,18,23)(15,24,20,25,26,16,27,28);(6,9)(10,12)(13,15)(14,16)(17,26)(18,27)(19,20)(21,24)(22,28)(23,25);(2,4)(5,7,8,11)(6,10,9,12)(13,14,17,21)(15,25,26,28)(16,27,24,20)(18,23,19,22);(7,11)(10,12)(13,16,19,28,17,24,18,25)(14,20,22,26,21,27,23,15)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288703);