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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,26,22,27,20,25)(19,23,28,24,30,32,29,31);(1,2)(3,4)(10,13)(14,15)(17,19,20,29,22,30,21,28)(18,23,25,31,27,32,26,24);(2,4)(17,19,22,30)(18,24,27,31)(20,28,21,29)(23,25,32,26);(9,11,12,16)(10,14,13,15)(17,20,22,21)(18,25,27,26)(19,28,30,29)(23,24,32,31)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267603);