Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13523 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2092 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,18,22,34)(8,23,25,36)(9,19)(10,16)(11,32)(12,17)(13,20,26,14)(15,29)(21,27,31,24)(28,33)(30,35);(1,2)(3,4)(5,7,9,20,11,21,29,36,12,22,30,14,10,31,28,23)(6,13,15,18,16,8,19,24,17,26,33,34,32,25,35,27);(2,4)(5,8)(6,14,17,20)(7,19,22,35)(9,27)(10,13)(11,26)(12,25)(15,21,33,31)(16,23,32,36)(18,29)(24,30)(28,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=eh,d^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322131, #72322145, #72322148, … |
| maximal normal subgroup of | #72322131, #72322145, #72322148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56428);