Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9471 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,8)(9,10,13,16,14,17,12,22)(11,18,21,23,19,24,20,15);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15,17,23)(12,20)(13,21)(14,19)(16,18,22,24);(1,3)(2,5)(4,7)(6,8)(11,19)(15,23)(18,24)(20,21)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ch,e^b=ei,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238108, #72238115 |
| maximal normal subgroup of | #72238108, #72238115 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15391);