Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384166 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14373 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1093 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)(3,11,14,7)(4,16,18,8)(5,19)(6,9)(10,20)(12,21,15,23)(13,24)(17,25,30,29)(22,27)(26,31)(28,32);(1,3)(2,7)(4,13)(5,12)(6,14)(8,24)(9,11)(10,21)(15,20)(16,22)(17,26)(18,27)(19,23)(25,32)(28,30)(29,31);(3,12,14,15)(7,21,11,23)(13,26,27,28)(22,31,24,32);(1,4)(2,8)(3,13)(5,17)(6,18)(7,22)(9,16)(10,25)(11,24)(12,26)(14,27)(15,28)(19,29)(20,30)(21,31)(23,32)Polycyclic presentation:
2^9|a^2=h,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285925, #72286050, #72286134, … |
| maximal normal subgroup of | #72285925, #72286050, #72286134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291362);