Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1958 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)(13,14,18,23)(15,25,26,20)(16,27,28,21)(17,22,19,24);(1,2,3,4)(13,15,18,26)(14,20,23,25)(16,19,28,17)(21,24,27,22);(6,9)(10,12)(13,15,19,16,18,26,17,28)(14,20,24,21,23,25,22,27);(13,16,19,26,18,28,17,15)(14,21,24,25,23,27,22,20);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,23,24)(15,16,26,28)(20,21,25,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374899);