Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,8)(13,14,17,19)(15,22,24,28)(16,20,23,21)(18,25,27,26);(1,2,3,4)(9,10)(11,12)(13,15,23,25,17,24,16,26)(14,18,20,28,19,27,21,22);(5,6,7,8)(10,12)(14,19)(15,24)(20,21)(25,26);(2,4)(14,20)(15,25)(16,23)(18,27)(19,21)(24,26)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^d=ei,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72264150, #72264168, #72264335, … |
| maximal normal subgroup of | #72264150, #72264168, #72264335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330049);