Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2474 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)(5,6,9,16,11,14,10,15)(7,12,19,28,18,29,25,27)(8,20,23,30,24,31,22,13)(17,32,34,36,21,35,33,26);(2,4)(5,7,10,19,11,18,9,25)(6,12,16,27,14,29,15,28)(8,21,23,34,24,17,22,33)(13,26,31,32,30,36,20,35);(5,8)(6,13)(7,17)(9,23)(10,22)(11,24)(12,26)(14,30)(15,31)(16,20)(18,21)(19,34)(25,33)(27,35)(28,32)(29,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265200, #72265207, #72265227, … |
| maximal normal subgroup of | #72265200, #72265207, #72265227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30843);