Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2270 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)(13,14,17,21,16,18,25,20)(15,22,24,27,23,28,26,19);(5,6)(7,8)(9,11,12,10)(14,18)(19,27)(20,21)(22,28);(2,4)(6,8)(10,11)(13,15,16,23)(14,19,18,27)(17,26,25,24)(20,22,21,28);(14,20)(17,25)(18,21)(19,22)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=efi,b^2=ei,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264878, #72264970, #72265165, … |
| maximal normal subgroup of | #72264878, #72264970, #72265165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302913);