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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(14,19)(15,22)(18,21)(20,26)(23,24)(25,28);(1,2)(3,4)(6,9)(10,12)(13,14)(15,23)(16,18)(17,19)(20,25)(21,27)(22,26)(24,28);(2,4)(13,15)(14,20)(16,26)(17,25)(18,28)(19,23)(21,22)(24,27);(2,4)(13,16)(14,21)(15,24)(17,27)(18,19)(20,28)(22,23)(25,26);(5,7,8,11)(6,10,9,12)(14,19)(16,27)(20,23)(24,26)Polycyclic presentation:
2^9|e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,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, 6347397);