Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| 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×#537 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(10,14)(12,20)(15,21)(16,22)(17,23)(19,24);(9,10,12,14,13,17,20,23)(11,16,19,22,18,15,24,21);(2,4)(9,11)(10,15)(12,19)(13,18)(14,21)(16,17)(20,24)(22,23);(5,7,8,6)(9,11)(10,16)(12,19)(13,18)(14,22)(15,17)(20,24)(21,23);(10,17)(11,18)(14,23)(19,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,e^b=ei,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72284435, #72284669, #72285114 |
| maximal normal subgroup of | #72284435, #72284669, #72285114 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280084);