Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140211 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(6,12)(7,17,20,36)(8,10)(9,23)(13,30,34,35)(14,15)(16,27)(18,28,33,32)(19,31,21,25)(22,26);(1,2,3,4)(5,6,8,12,11,16,10,27)(7,18,19,30,20,33,21,35)(9,24,22,14,26,29,23,15)(13,31,32,17,34,25,28,36);(5,7)(6,13)(8,21)(9,25)(10,19)(11,20)(12,28)(14,30)(15,35)(16,34)(17,22)(18,29)(23,36)(24,33)(26,31)(27,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239802, #72239813, #72239826, … |
| maximal normal subgroup of | #72239802, #72239813, #72239826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55356);