Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | |
2626 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);(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,21,22,28)(16,24,26,19)(18,25,27,23);(1,2,3,4)(13,15)(14,18)(16,25)(17,22)(19,28)(20,27)(21,24)(23,26);(5,7,8,11)(6,10,9,12)(15,22)(18,27)(21,28)(23,25);(6,9)(10,12)(14,19)(16,26)(18,28)(20,24)(21,27)(23,25)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282839, #72283133, #72283430, … |
| maximal normal subgroup of | #72282839, #72283133, #72283430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590724);