Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6067 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)(8,14)(9,13)(10,16)(11,15)(12,18)(19,33)(20,27)(21,32)(22,26)(23,28)(24,30)(25,31)(29,34)(35,36);(2,4)(5,7)(6,12)(8,22)(9,20)(10,18)(11,21)(13,17)(14,29)(15,27)(16,32)(19,23)(24,35)(25,34)(26,28)(30,33)(31,36);(6,13)(7,18)(9,10)(14,28)(15,16)(17,32)(19,22)(20,21)(23,25)(29,33)(30,31)(34,35);(5,8)(6,14)(7,19)(9,23)(10,25)(11,24)(12,26)(13,28)(15,30)(16,31)(17,33)(18,22)(20,35)(21,34)(27,36)(29,32);(6,15)(12,27)(13,16)(14,30)(17,32)(26,36)(28,31)(29,33)Polycyclic presentation:
2^9|f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237505, #72256284, #72256291, … |
| maximal normal subgroup of | #72237505, #72256284, #72256291, … |
| maximal quotient of | #72310284, #72310288, #72310305, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344798);