Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,4,3)(13,14)(15,21)(16,24)(17,20)(18,22)(19,25)(23,27)(26,28);(5,6,7,10,8,9,11,12)(13,15)(14,18,20,26)(16,22)(17,23)(19,21,24,27)(25,28);(1,3,4,2)(6,9)(10,12)(16,25)(19,24)(21,27)(22,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ci,e^b=eh,d^c=dh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239304, #72239307, #72239308, … |
| maximal normal subgroup of | #72239304, #72239307, #72239308, … |
| maximal quotient of | #72239316, #72239342 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16687);