Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #385935 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14417 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1429 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)(3,14)(4,10,15,27)(5,22,24,31)(6,25)(7,12)(8,13)(9,16)(11,23,26,19)(17,32)(18,28)(20,30,21,29);(1,3)(2,9)(4,19)(5,21)(6,18)(7,16)(8,17)(10,22)(11,30)(12,32)(13,28)(14,25)(15,23)(20,24)(26,29)(27,31);(1,4)(2,10,13,27)(3,5)(6,21)(7,20)(8,15)(9,26,28,11)(12,30,25,29)(14,31,32,22)(16,19)(17,24)(18,23);(1,5,8,24)(2,11,13,26)(3,15,17,4)(6,19,7,23)(9,27,28,10)(12,22,25,31)(14,29,32,30)(16,20,18,21)Polycyclic presentation:
2^9|a^2=h,c^2=eg,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215912, #72216181, #72216203, … |
| maximal normal subgroup of | #72215912, #72216181, #72216203, … |
| maximal quotient of | C2×#385936, #72252154, #72286836 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293132);