Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2031 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,21,18,22,26,19)(15,20,24,25,23,28,27,16);(5,7,8,11)(6,9,10,12)(14,19)(16,20)(17,26)(21,22)(24,27)(25,28);(6,10)(9,12)(13,15,18,23)(14,20,22,28)(16,21,25,19)(17,24,26,27);(13,16,17,20,18,25,26,28)(14,15,21,24,22,23,19,27);(15,23)(16,25)(20,28)(24,27)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^c=ei,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287071);