Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11753 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,22)(16,20)(17,25)(18,21)(19,23)(24,27)(26,28);(6,9)(7,12)(10,11)(13,15)(14,19,21,28)(16,23,17,26)(18,24)(20,27)(22,25);(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,28,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=g,b^a=bd,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh,h^b=hi,f^c=fi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282585, #72282618, #72282631, … |
| maximal normal subgroup of | #72282585, #72282618, #72282631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34812);