Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,12,16,10,14,28,17)(7,18,21,24,22,34,29,33)(8,23,15,11,25,9,26,27)(13,20,30,19,32,35,31,36);(2,4)(5,7)(6,13)(8,24)(9,20)(10,22)(11,19)(12,29)(14,32)(15,18)(16,31)(17,30)(21,28)(23,35)(25,33)(26,34)(27,36);(5,8)(6,9)(7,13)(10,25)(11,17)(12,15)(14,23)(16,27)(18,20)(19,24)(21,30)(22,32)(26,28)(29,31)(33,36)(34,35)Polycyclic presentation:
2^9|a^2=g,d^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,e^b=ei,f^b=fh,g^b=gi,h^b=hi,d^c=dhi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55137);