Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | C2×C4×D4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6,7,9)(8,10,12,11)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(1,2,4,3)(6,9)(10,11)(13,15,16,22,17,21,23,27)(14,18,19,25,20,24,26,28);(1,3,4,2)(6,10)(8,12)(9,11);(15,21)(18,24)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=di,f^c=fh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72343945, #72343948, #72343963 |
| maximal normal subgroup of | #72343945, #72343948, #72343963 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109574);