Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| 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 | C22×#469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2230 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)(13,14,19,24)(15,25,27,21)(16,28,26,20)(17,23,18,22);(5,6)(7,8)(13,15,18,16,19,27,17,26)(14,20,23,21,24,28,22,25);(2,4)(6,8)(15,16)(17,18)(20,21)(22,23)(25,28)(26,27);(10,12)(13,16,18,27,19,26,17,15)(14,21,23,28,24,25,22,20)Polycyclic presentation:
2^9|a^2=i,b^2=ei,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264150, #72264195, #72264314, … |
| maximal normal subgroup of | #72264150, #72264195, #72264314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361583);