Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #99 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C22×#95 | C22 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25 ◁ C22×#95 ◁ G | |
| LECS | G ▷ C22×#95 ▷ C24 ▷ C2 ▷ C1 | |
954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(22,25)(26,27);(6,9,11,10)(13,14,15,17,16,18,19,20)(21,22)(23,27,28,25)(24,26)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,e^2=h,b^a=bc,c^a=cf,e^a=efg,c^b=cg,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238061, #72267754, #72267773, … |
| maximal normal subgroup of | #72238061, #72267754, #72267773, … |
| maximal quotient of | #72238066, #72238067, #72238068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 568);