Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,8)(13,14,19,15,17,21,18,22)(16,23,27,20,25,28,26,24);(1,2,4,3)(9,10,11,12)(13,15,19,21,17,22,18,14)(16,24,27,23,25,20,26,28);(6,8)(10,12)(13,16,17,25)(14,20,21,24)(15,23,22,28)(18,27,19,26);(1,3,4,2)(14,21)(15,22)(20,24)(23,28)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=i,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264858, #72264877, #72264943, … |
| maximal normal subgroup of | #72264858, #72264877, #72264943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269248);