Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19671 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#127 |
| prime radical | #19619 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,14,8)(3,11,4,7,12,15,13,16)(17,18,21,24)(19,26,28,23)(20,30,25,22)(27,32,31,29);(1,3,5,4,6,12,14,13)(2,7,8,11,10,16,9,15)(17,19,21,28)(18,23,24,26)(20,31,25,27)(22,29,30,32);(17,20,22,18,21,25,30,24)(19,27,29,26,28,31,32,23);(1,4,5,12,6,13,14,3)(2,7,9,15,10,16,8,11)(19,28)(23,26)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=gi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ci,d^b=dhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285558, #72322026, #72322056 |
| maximal normal subgroup of | #72285558, #72322026, #72322056 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120535);