Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2633 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,25)(14,19,21,23)(16,26,22,28)(18,27,20,24);(1,2,3,4)(13,16)(14,20)(15,19)(17,22)(18,21)(23,25)(24,28)(26,27);(5,7,8,11)(6,10,9,12)(14,21)(16,22)(19,23)(26,28);(6,9)(10,12)(14,21)(15,24)(18,20)(19,28)(23,26)(25,27)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72282700, #72283056, #72283220, … |
| maximal normal subgroup of | #72282700, #72283056, #72283220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590716);