Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,13,15,10)(8,17)(9,11)(12,18)(14,19,16,20)(21,22)(23,29,31,35)(24,27)(25,28)(26,30,34,32)(33,36);(5,7,9,15)(6,10,11,13)(8,14,18,16)(12,19,17,20)(21,23)(22,26)(24,32)(25,31)(27,35)(28,34)(29,36)(30,33);(2,4)(7,14)(8,18)(10,19)(12,17)(13,20)(15,16)(26,34)(29,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,f^c=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72240283, #72240337, #72240340, … |
| maximal normal subgroup of | #72240283, #72240337, #72240340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20163);