Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141926 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1993 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
498 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)(7,18,21,36)(8,15)(9,13)(10,28)(11,16)(12,19,31,26)(14,24)(17,27)(20,35,23,32)(22,33,29,30)(25,34);(1,2)(3,4)(5,7)(6,12,16,31)(8,23)(9,26)(10,29)(11,21)(13,33,14,30)(15,18,34,36)(17,35,28,32)(19,24)(20,25)(22,27);(2,4)(6,13,16,14)(8,24)(9,25)(10,27)(12,30,31,33)(15,28,34,17)(18,35,36,32)(19,23)(20,26)(22,29)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=hi,h^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=dh,e^b=ehi,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240725, #72240728, #72240731, … |
| maximal normal subgroup of | #72240725, #72240728, #72240731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56533);