Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
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 | C2×#3058 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2818 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)(5,6,8,7)(9,10,14,18,15,19,13,20)(11,16,21,23,17,22,24,12);(5,7,8,6)(9,11)(10,12)(13,21)(14,24)(15,17)(16,20)(18,22)(19,23);(2,4)(9,11)(10,16)(12,20)(13,24)(14,21)(15,17)(18,23)(19,22);(9,12,14,16,15,23,13,22)(10,17,18,24,19,11,20,21);(9,13,15,14)(10,18,19,20)(11,21,17,24)(12,22,23,16)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,d^b=df,f^b=fi,h^b=hi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72214822, #72280229, #72280494, … |
| maximal normal subgroup of | #72214822, #72280229, #72280494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289171);