Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #364600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13967 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2407 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)(3,13,16,8,18,25,17,26)(4,19,20,9,22,27,21,28)(14,29,30,32,31,23,15,24);(1,3,6,16,7,18,5,17)(2,8,10,25,12,26,11,13)(4,15,20,14,22,30,21,31)(9,24,27,29,28,32,19,23);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,23)(10,19)(11,27)(12,28)(13,24)(15,16)(17,30)(18,31)(25,32)(26,29);(1,4)(2,9)(3,15)(5,21)(6,20)(7,22)(8,24)(10,27)(11,19)(12,28)(13,23)(14,16)(17,31)(18,30)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215972, #72216029, #72216112, … |
| maximal normal subgroup of | #72215972, #72216029, #72216112, … |
| maximal quotient of | #72237506, #72237507, C2×#364602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271798);