Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433042 |
| 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 | #34633 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2118 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)(7,11)(10,12)(13,14,19,22)(15,24,26,28)(16,25,20,21)(17,23,18,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,25,19,26,17,21)(14,20,23,24,22,16,27,28);(2,4)(6,9)(10,12)(13,16)(14,21)(15,23)(17,24)(18,28)(19,20)(22,25)(26,27);(5,7,8,11)(6,10,9,12)(14,22)(16,20)(23,27)(24,28);(13,17,19,18)(14,23,22,27)(15,25,26,21)(16,24,20,28)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342746);