Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | C4×#120 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3371 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)(5,6,11,16,12,17,10,19)(7,18,14,13,22,24,21,8)(9,15,25,31,26,32,27,30)(20,34,29,23,36,33,35,28);(2,4)(5,7,11,14,12,22,10,21)(6,13,16,24,17,8,19,18)(9,20,25,29,26,36,27,35)(15,28,31,34,32,23,30,33);(5,8)(6,14)(7,19)(9,23)(10,24)(11,18)(12,13)(15,29)(16,22)(17,21)(20,30)(25,33)(26,28)(27,34)(31,36)(32,35);(5,9)(6,15)(7,20)(8,23)(10,27)(11,25)(12,26)(13,28)(14,29)(16,31)(17,32)(18,33)(19,30)(21,35)(22,36)(24,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,g^b=gi,f^c=fi,g^c=gi,e^d=ei,f^d=fi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72250213, #72250285, #72257145, … |
| maximal normal subgroup of | #72250213, #72250285, #72257145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299798);