Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13490 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2073 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,16)(7,18)(8,12,9,24)(11,27,23,17)(13,26)(14,25,15,22)(19,31)(20,28)(21,29)(30,35)(32,34)(33,36);(1,2)(3,4)(6,12)(7,19)(8,22)(9,25)(11,23)(14,17)(15,27)(16,24)(18,32)(20,36)(21,26)(28,29)(30,33)(34,35);(2,4)(5,7,8,19,23,34,25,36,10,21,9,26,11,20,22,35)(6,13,14,29,27,33,24,32,16,31,15,18,17,30,12,28)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=hi,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,h^b=hi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239853, #72239860, #72240845, … |
| maximal normal subgroup of | #72239853, #72239860, #72240845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55712);