Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2538 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)(13,14,16,21,17,20,25,18)(15,22,24,26,23,19,27,28);(2,3)(14,18)(16,25)(19,26)(20,21)(22,28)(24,27);(1,2,4,3)(6,9)(10,12)(13,15)(14,19)(16,24)(17,23)(18,26)(20,22)(21,28)(25,27);(1,3,4,2)(14,20)(18,21)(19,22)(26,28);(5,7,8,11)(6,10,9,12)(15,23)(19,22)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374598);