Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1964 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,16)(7,9)(8,19,22,35)(10,25,26,15)(12,14)(13,28,30,21)(17,32)(18,27)(20,33)(23,34)(24,31)(29,36);(2,4)(5,7,12,16,11,14,9,6)(8,17,23,30,22,29,20,13)(10,18,27,25,26,31,24,15)(19,21,33,36,35,28,34,32);(5,8,10,21,11,22,26,28)(6,13,15,19,16,30,25,35)(7,17,18,33,14,29,31,34)(9,20,24,32,12,23,27,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,h^a=hi,d^b=dh,f^b=fh,h^b=hi,f^d=fi,g^d=gi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311847, #72311849, #72311871, … |
| maximal normal subgroup of | #72311847, #72311849, #72311871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54273);