Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| 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 |
| prime radical | #35513 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2859 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)(5,6)(7,11)(8,10)(9,12)(14,19)(20,28)(21,24)(22,23);(5,7)(6,9)(8,12)(10,11)(13,14,17,19)(15,24,26,21)(16,23,18,22)(20,27,28,25);(2,4)(13,15)(14,20)(16,25)(17,26)(18,27)(19,28)(21,23)(22,24);(2,4)(6,10)(7,12)(13,15)(14,21)(16,27)(17,26)(18,25)(19,24)(20,23)(22,28);(13,16,17,18)(14,22,19,23)(15,25,26,27)(20,24,28,21)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410350);