Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(14,19)(20,27)(21,25)(22,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,22)(16,21)(17,19)(18,25)(20,23)(24,28)(26,27);(2,4)(13,15)(14,20)(16,23)(17,24)(18,26)(19,27)(21,28)(22,25);(2,4)(6,10)(7,12)(14,19)(20,27)(21,25)(22,28);(13,16,17,18)(14,21,19,25)(15,23,24,26)(20,28,27,22)Polycyclic presentation:
2^9|a^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410308);