Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#121 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22)(9,15)(10,16)(11,14)(12,20)(13,25)(18,33)(19,28)(21,29)(23,30)(24,31)(26,32)(27,35)(34,36);(2,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,28)(15,17)(16,29)(18,24)(22,33)(25,35)(26,34)(27,32)(31,36);(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,31)(15,22)(16,32)(17,30)(19,34)(20,23)(21,35)(28,33)(29,36);(6,14)(12,28)(13,31)(15,16)(17,29)(22,32)(27,33)(30,36);(5,9,11,10)(6,15,14,16)(7,19,21,20)(8,24,26,25)(12,17,28,29)(13,22,31,32)(18,34,35,23)(27,30,33,36)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,h^a=hi,c^b=ch,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237503, #72237505, #72250753, … |
| maximal normal subgroup of | #72237503, #72237505, #72250753, … |
| maximal quotient of | #72310323, C2×#6431784 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338980);