Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #34936 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,16,18,17,21,25,20)(15,22,24,28,23,19,27,26);(5,7,11,6,9,12,8,10)(14,18,21,20)(15,23)(19,28,22,26)(24,27);(1,2,3,4)(6,10)(7,12)(13,15)(14,19)(16,24)(17,23)(18,26)(20,28)(21,22)(25,27);(15,23)(19,22)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ch,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72215872, #72216042, #72216054, … |
| maximal normal subgroup of | #72215872, #72216042, #72216054, … |
| maximal quotient of | #72321602 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111336);