Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2424 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)(6,9)(7,11)(13,14,19,23)(15,17,25,20)(16,26,27,21)(18,22,24,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,21)(17,23)(18,28)(19,25)(22,24)(26,27);(1,2,3,4)(13,16,19,27)(14,21,23,26)(15,24,25,18)(17,22,20,28);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,15,23,25)(16,22,27,28)(18,21,24,26);(6,9)(10,12)(13,18)(14,22)(15,16)(17,26)(19,24)(20,21)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590982);