Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4716 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)(9,10,14,11,13,15,21,17)(12,16,20,18,19,22,24,23);(2,4)(9,11,14,15,13,17,21,10)(12,18,20,22,19,23,24,16);(6,8)(10,15)(11,17)(16,22)(18,23);(9,12)(10,16)(11,18)(13,19)(14,20)(15,22)(17,23)(21,24);(12,19)(16,22)(18,23)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,h^2=i,b^a=bf,c^a=cg,c^b=ci,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72316572, #72316579, #72316600 |
| maximal normal subgroup of | #72316572, #72316579, #72316600 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278539);