Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,28)(25,27);(2,3)(6,9)(10,11)(13,15,17,24,18,23,16,25)(14,19,20,22,21,27,26,28);(1,2,4,3)(5,7,8,12)(6,10,9,11)(15,23)(19,27)(22,28)(24,25);(1,3,4,2)(15,23)(19,27)(22,28)(24,25);(13,16,18,17)(14,20,21,26)(15,24,23,25)(19,28,27,22)Polycyclic presentation:
2^9|b^2=fg,c^2=ghi,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373731);