Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, prime radical, cosocle | C25 | C24 |
| socle | C23 | C22×C22⋊C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,21,23,18)(16,19)(17,20)(22,27,28,25)(24,26);(1,2,3,4)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(6,9)(10,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=f,b^a=be,c^a=cf,e^a=ei,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72240819, #72249970, #72249990, … |
| maximal normal subgroup of | #72240819, #72249970, #72249990, … |
| maximal quotient of | #72240802, #72240803, #72264520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260994);