Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1013 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)(7,11)(8,12)(9,10)(13,14,17,22,18,23,27,21)(15,19,16,25,24,20,26,28);(6,10)(11,12)(13,15,17,16,18,24,27,26)(14,19,22,25,23,20,21,28);(5,7,9,8)(6,11,10,12)(13,16,17,24,18,26,27,15)(14,20,21,25,23,19,22,28);(2,4)(5,8,9,7)(6,12,10,11)(15,24)(16,26)(19,20)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ef,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,c^b=ci,d^b=di,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300486);