Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4342 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)(5,6,10,14,11,16,9,15)(7,17,19,12,20,27,25,28)(8,13,24,31,22,30,23,32)(18,29,34,26,33,36,21,35);(5,7)(6,12)(8,21)(9,25)(10,19)(11,20)(13,29)(14,27)(15,17)(16,28)(18,24)(22,34)(23,33)(26,31)(30,36)(32,35);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,22)(12,26)(14,31)(15,32)(16,30)(17,29)(19,34)(20,33)(21,25)(27,36)(28,35);(8,22)(13,30)(18,33)(21,34)(23,24)(26,35)(29,36)(31,32)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,g^b=gi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237177, #72237180, #72237188, … |
| maximal normal subgroup of | #72237177, #72237180, #72237188, … |
| maximal quotient of | #72239062, #72257449, #72257487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264771);