Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2408 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13991 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,13,15,8,17,24,16,25)(4,18,21,9,22,27,20,28)(14,30,19,29,31,23,32,26);(1,3)(2,8)(4,19,22,32)(5,16)(6,15)(7,17)(9,26,28,29)(10,24)(11,13)(12,25)(14,21,31,20)(18,23,27,30);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,23)(10,27)(11,18)(12,28)(13,29)(15,19)(16,32)(17,31)(24,26)(25,30);(1,3)(2,8)(4,14)(5,16)(6,15)(7,17)(9,23)(10,24)(11,13)(12,25)(18,29)(19,21)(20,32)(22,31)(26,27)(28,30)Polycyclic presentation:
2^9|a^2=h,b^2=gh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286635, #72287308, #72287334, … |
| maximal normal subgroup of | #72286635, #72287308, #72287334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273562);