Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| 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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,22)(18,23)(19,24)(25,28)(26,27);(1,2,3,4)(13,15,17,24)(14,19,22,20)(16,18,25,26)(21,27,28,23);(6,9)(10,12)(13,15,18,16,17,24,26,25)(14,20,23,21,22,19,27,28);(13,16,18,24,17,25,26,15)(14,21,23,19,22,28,27,20);(5,7,8,11)(6,10,9,12)(14,22)(19,20)(21,28)(23,27)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375068);