Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(15,22)(16,17)(19,26)(20,24)(23,25)(27,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,24)(17,20)(18,21)(22,26)(23,28)(25,27);(2,4)(13,15,18,23)(14,19,21,28)(16,22,17,25)(20,27,24,26);(2,4)(13,14)(15,19)(16,24)(17,20)(18,21)(22,26)(23,28)(25,27);(13,16,18,17)(14,20,21,24)(15,22,23,25)(19,27,28,26)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345345);