Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| 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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,32)(29,31);(2,4)(5,7,8,6)(17,19,26,29,20,27,21,28)(18,22,30,32,24,23,25,31);(10,13)(14,16)(17,19,20,27)(18,23,24,22)(21,29,26,28)(25,31,30,32);(2,4)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=h,b^2=egh,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274881);