Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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×#3269 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2018 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)(7,10)(8,9)(11,12)(13,14,18,19)(15,23,25,20)(16,21,26,28)(17,22,27,24);(1,2,4,3)(6,9)(10,12)(14,19)(15,20)(17,22)(21,28)(23,25)(24,27);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,17)(18,25)(19,23)(21,24)(22,28)(26,27);(13,16,18,26)(14,21,19,28)(15,24,25,22)(17,20,27,23);(13,17)(14,22)(15,16)(18,27)(19,24)(20,28)(21,23)(25,26)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=fi,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486651);