Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454029 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,18)(9,21)(10,13)(12,19)(14,23)(15,29)(16,24)(17,33)(20,28)(22,27)(25,31)(26,32)(30,36)(34,35);(2,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,25)(13,24)(14,26)(18,29)(19,31)(20,30)(21,33)(22,34)(23,32)(27,35)(28,36);(1,2)(3,4)(11,24)(25,35)(26,33)(29,36);(2,4)(5,8,10,20)(6,12,13,27)(7,15,16,30)(9,22,23,19)(11,25,24,35)(14,28,21,18)(17,34,32,31)(26,36,33,29)Polycyclic presentation:
2^9|d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72287194, #72313084, #72313106, … |
| maximal normal subgroup of | #72287194, #72313084, #72313106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361246);