Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2375 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)(5,6,9,11)(7,12,8,10);(5,7,9,8)(6,10,11,12)(13,14)(15,22)(16,19)(17,20)(18,24)(21,26)(23,28)(25,27);(2,4)(5,8,9,7)(6,10,11,12)(13,15)(14,18)(16,25)(17,23)(19,22)(20,27)(21,28)(24,26);(14,19)(16,26)(18,22)(20,21)(24,25)(27,28);(14,20)(15,23)(19,21)(24,25)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72256834, #72256996, #72257000, … |
| maximal normal subgroup of | #72256834, #72256996, #72257000, … |
| maximal quotient of | #72308647 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346321);