Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(6,10)(7,8)(11,12)(13,14)(15,21,23,27)(16,19)(17,20)(18,22,25,28)(24,26);(5,6,8,12,9,11,7,10)(13,15)(14,18,20,25)(16,22)(17,23)(19,21,26,27)(24,28);(1,2,3,4)(6,11)(10,12)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=hi,f^2=h,b^a=bd,d^a=dh,f^a=fhi,c^b=ch,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240740, #72257884, #72257926, … |
| maximal normal subgroup of | #72240740, #72257884, #72257926, … |
| maximal quotient of | #72240741, #72240743 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12591);