Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,8,12,10)(14,18)(15,19)(16,23)(20,24)(21,27)(22,25);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(13,14,17,18)(15,19,22,25)(16,20,24,23)(21,26,27,28);(2,4)(13,15)(14,19)(16,21)(17,22)(18,25)(20,26)(23,28)(24,27);(5,8,9,10)(6,7,11,12)(13,16,17,24)(14,20,18,23)(15,21,22,27)(19,26,25,28);(15,22)(19,25)(21,27)(26,28)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^c=ei,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409675);