Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(15,22)(19,27)(23,28)(24,25);(6,9)(7,11)(13,14)(15,23)(16,20)(17,21)(18,26)(19,25)(22,28)(24,27);(1,2,3,4)(13,15,16,24,17,22,18,25)(14,19,26,28,21,27,20,23);(6,9)(10,12)(13,16,17,18)(14,20,21,26)(15,24,22,25)(19,23,27,28);(5,7,8,11)(6,10,9,12)(13,15,16,24,17,22,18,25)(14,19,26,28,21,27,20,23)Polycyclic presentation:
2^9|c^2=gh,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723063);