Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #385 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
776 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)(9,12)(13,14,17,23,19,24,18,22)(15,21,16,27,26,28,25,20);(5,6)(7,9)(8,10)(11,12)(13,15,18,25,19,26,17,16)(14,20,22,28,24,27,23,21);(1,3,4,2)(5,7)(6,9)(8,11)(10,12)(13,16,17,26,19,25,18,15)(14,21,23,27,24,28,22,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dh,e^b=ei,d^c=di,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17154);