Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C23×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1820 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
641 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)(7,17,22,32)(8,23)(9,25,26,14)(10,27,19,15)(11,16)(12,20)(13,24,30,34)(18,29)(21,31)(28,33)(35,36);(1,2)(3,4)(5,7,18,34,11,22,35,24)(6,12,29,33,16,23,36,31)(8,19,21,26,20,10,28,9)(13,14,32,27,30,25,17,15);(2,4)(5,8,18,21,11,20,35,28)(6,13,29,32,16,30,36,17)(7,19,34,26,22,10,24,9)(12,25,33,15,23,14,31,27)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,g^c=gh,g^d=gi,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56517);