Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #379827 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14255 |
| 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 | |
2112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,8,5,11)(3,9,13,24,15,20,19,22)(4,10,18,27,16,21,17,26)(14,23,28,32,29,25,30,31);(2,8)(3,13)(5,6)(9,22)(10,21)(14,28)(15,19)(17,18)(20,24)(23,31)(25,32)(29,30);(1,3)(2,9)(4,14)(5,19)(6,13)(7,15)(8,20)(10,25)(11,22)(12,24)(16,29)(17,30)(18,28)(21,23)(26,32)(27,31);(1,4)(2,10)(3,14)(5,17)(6,18)(7,16)(8,21)(9,23)(11,26)(12,27)(13,28)(15,29)(19,30)(20,25)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72318290, #72318474, #72318482, … |
| maximal normal subgroup of | #72318290, #72318474, #72318482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287025);