Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator | C22 | C23×C42 |
| socle | C23 | C24×C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(14,18)(15,21)(20,25)(22,27);(5,7)(6,9)(8,12)(10,11)(13,14,17,20,16,18,23,25)(15,19,22,26,21,24,27,28);(7,12)(9,11)(13,15)(14,19)(16,21)(17,22)(18,24)(20,26)(23,27)(25,28);(6,10)(9,11)(15,21)(19,24)(22,27)(26,28);(1,3,4,2)(14,18)(19,24)(20,25)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=g,h^2=i,b^a=bf,c^a=ci,d^a=dfi,c^b=cfi,e^b=ei,d^c=di152 conjugacy classes — view the full character table.
| maximal subgroup of | #72312024, #72312068, #72312075, … |
| maximal normal subgroup of | #72312024, #72312068, #72312075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277580);