Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #44877 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(7,11)(10,12)(13,14,20,23,19,15,18,17)(16,21,28,24,25,22,26,27);(6,9)(7,11)(13,15,19,14)(16,22,25,21)(17,20,23,18)(24,26,27,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,22)(17,27)(18,26)(19,25)(20,28)(23,24);(6,9)(10,12)(13,17,18,15,19,23,20,14)(16,24,26,21,25,27,28,22);(5,7,8,11)(6,10,9,12)(16,25)(21,22)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283437, #72283488, #72284220, … |
| maximal normal subgroup of | #72283437, #72283488, #72284220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7052982);