Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#449 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(9,10)(11,12)(14,19)(15,23)(16,17)(20,28)(21,22)(25,26);(5,6)(7,8)(13,14)(15,24)(16,21)(17,19)(18,22)(20,25)(23,28)(26,27);(2,4)(6,8)(13,15,16,23,18,26,17,25)(14,20,19,27,22,28,21,24);(10,12)(13,16,18,17)(14,21,22,19)(15,23,26,25)(20,24,28,27)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264572, #72264575, #72264581, … |
| maximal normal subgroup of | #72264572, #72264575, #72264581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301809);