Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
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×C4⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(10,13)(11,15)(17,18,22,26,21,25,32,27)(19,20,29,31,28,23,24,30);(1,3)(2,4)(5,8)(6,7)(9,10,12,13)(11,14,15,16)(17,19,21,28)(18,23,25,20)(22,29,32,24)(26,30,27,31);(1,4,6,5)(2,7,8,3)(9,11,12,15)(10,14,13,16)(17,20,21,23)(18,24,25,29)(19,27,28,26)(22,31,32,30);(1,5,6,4)(2,7,8,3)(10,13)(14,16)(18,25)(19,28)(24,29)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292305);