Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,6,8,5,10)(3,7,11,15,13,14,12,16)(17,18)(19,23)(20,24,28,22)(21,25)(26,30,32,29)(27,31);(1,3,5,12,6,13,4,11)(2,7,10,16,8,14,9,15)(18,22,25,24)(23,29,31,30);(2,8)(7,14)(9,10)(15,16)(17,19,21,27)(18,23,25,31)(20,26,28,32)(22,29,24,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=i,e^2=h,f^2=h,b^a=bd,c^a=ch,d^a=dg,f^a=fi,g^a=gi,e^b=egi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72316723, #72316725, #72316733, … |
| maximal normal subgroup of | #72316723, #72316725, #72316733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34322);