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 | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6)(7,12,8,21)(9,17)(10,25)(11,15)(13,22,33,19)(14,27)(16,18)(20,30,23,31)(24,28)(26,34,36,29)(32,35);(1,2,4,3)(5,7,18,23,9,8,11,20)(6,12,16,31,17,21,15,30)(10,19,35,26,24,22,27,36)(13,32,34,28,33,14,29,25);(1,3,4,2)(5,8,9,7)(6,13)(10,26,24,36)(11,20,18,23)(12,28)(14,30)(15,34)(16,29)(17,33)(19,27,22,35)(21,25)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=d,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,g^b=gi,e^c=eh,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250660, #72250675, #72250684, … |
| maximal normal subgroup of | #72250660, #72250675, #72250684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55937);