Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #94871 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,8,12,9,11,10,7)(13,14)(15,22,24,28)(16,20)(17,26)(18,21)(19,23,27,25);(5,7,10,11,9,12,8,6)(13,15,17,25,18,24,16,23)(14,19,20,22,21,27,26,28);(1,2,3,4)(5,8,9,10)(6,7,11,12)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,g^a=gh,c^b=ci,f^b=fh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72237202, #72238780, #72238788, … |
| maximal normal subgroup of | #72237202, #72238780, #72238788, … |
| maximal quotient of | #72237205, #72237209 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15973);