Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(14,19)(15,23,25,26)(20,24,27,28)(21,22);(5,7,8,12)(6,9,10,11)(13,14,17,21,16,19,18,22)(15,24,23,20,25,28,26,27);(1,2,3,4)(13,15,17,26,16,25,18,23)(14,20,21,24,19,27,22,28);(6,10)(9,11)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=fh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72317273, #72317284, #72317285 |
| maximal normal subgroup of | #72317273, #72317284, #72317285 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282338);