Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
17096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(10,14)(12,19)(15,21)(16,20)(18,23)(22,24);(9,10,13,16)(11,15,17,22)(12,20,19,14)(18,24,23,21);(2,4);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,23)(20,24);(11,17)(15,22)(18,23)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72316527, #72316530, #72316605 |
| maximal normal subgroup of | #72316527, #72316530, #72316605 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277919);