Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #387433 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2372 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14476 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,14)(4,20,22,26)(5,10,21,27)(6,12)(7,11)(8,13)(9,17)(15,29,31,23)(16,32,30,25)(18,24)(19,28);(1,3,6,18,8,19,7,17)(2,9,12,14,13,24,11,28)(4,16,21,31,22,30,5,15)(10,25,20,29,27,32,26,23);(1,4,6,21,8,22,7,5)(2,10,11,26,13,27,12,20)(3,15,17,30,19,31,18,16)(9,23,14,25,24,29,28,32);(1,5,7,22,8,21,6,4)(2,10,11,26,13,27,12,20)(3,16,18,31,19,30,17,15)(9,23,14,25,24,29,28,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294724);