Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129627 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,17)(8,21,9,14)(10,15)(11,16)(12,18,29,23)(13,22,31,24)(19,27)(20,32)(25,30)(26,33)(28,34,36,35);(1,2)(3,4)(5,7,11,20)(6,12,16,29)(8,18,9,23)(10,25,26,19)(13,30,31,27)(14,32,21,17)(15,28,33,36)(22,35,24,34);(2,4)(5,8)(6,13)(7,18)(9,11)(10,24)(12,27)(14,15)(16,31)(17,28)(19,35)(20,23)(21,33)(22,26)(25,34)(29,30)(32,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,e^c=ei,g^c=gi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55030);