Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,24)(17,20)(18,23)(19,25)(22,27)(26,28);(13,15)(14,18,20,26)(16,23)(17,22)(19,21,24,27)(25,28);(6,9)(10,12);(5,7,8,11)(6,10,9,12)(15,22)(18,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=f,e^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72213056, #72216224, #72216619, … |
| maximal normal subgroup of | #72213056, #72216224, #72216619, … |
| maximal quotient of | #72213013, #72240803, #72268628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260978);