Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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 | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,3)(5,6,7,8)(10,13)(11,15)(17,18)(19,23)(20,25)(21,22)(24,27)(26,32)(28,31)(29,30);(9,10,12,13)(11,16,15,14)(18,22)(19,26)(20,27)(23,31)(28,29)(30,32);(1,2,4,3)(17,19,27,28,21,29,20,26)(18,23,24,31,22,30,25,32);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,20,21,27)(18,24,22,25)(19,28,29,26)(23,32,30,31)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=fgi,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294026);