Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13389 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2033 |
| 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 | |
716 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)(5,6,9,15,11,16,24,13)(7,17,21,33,22,29,36,34)(8,23,18,14,10,26,25,32)(12,19,28,27,31,35,30,20);(5,7,18,19,9,21,10,27,11,22,25,35,24,36,8,20)(6,12,23,34,13,30,32,29,16,31,26,33,15,28,14,17);(2,4)(6,13)(7,19)(8,18)(9,24)(10,25)(12,17)(14,32)(15,16)(20,21)(22,35)(27,36)(28,34)(29,31)(30,33)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,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,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239848, #72239855, #72239861, … |
| maximal normal subgroup of | #72239848, #72239855, #72239861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55605);