Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(14,19)(15,24)(16,17)(20,23)(21,28)(22,27)(25,26);(5,6,7,10,8,9,11,12)(13,14,15,20,17,22,25,27,18,23,26,19,16,21,24,28);(1,3,4,2)(6,9)(10,12)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,h^a=hi,c^b=cgh,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72318613, #72318636, #72318656, … |
| maximal normal subgroup of | #72318613, #72318636, #72318656, … |
| maximal quotient of | #72322438, #72322636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35786);