Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14)(15,23)(16,24)(17,27)(18,22)(19,21)(20,26)(25,28);(2,3)(5,7)(6,10)(8,11)(9,12)(13,15,18,26,19,25,16,17)(14,20,24,23,21,27,22,28);(1,2,4,3)(5,8,9,6)(7,11,12,10)(14,21)(15,25)(17,26)(22,24);(1,3,4,2)(7,12)(10,11)(13,16,19,18)(14,22,21,24)(15,26,25,17)(20,23,27,28);(13,17,16,25,19,26,18,15)(14,23,22,20,21,28,24,27)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=hi,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511623);