Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417686 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15376 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,23,31)(3,13,16,25,17,29,22,8)(4,9,18,24,15,27,19,32)(5,10,20,28,21,26,14,30);(1,3)(2,8)(4,14)(5,19)(6,22)(7,17)(9,26)(10,27)(11,29)(12,25)(13,31)(15,20)(16,23)(18,21)(24,28)(30,32);(1,4)(2,9)(3,14)(5,16)(6,18)(7,15)(8,10)(11,24)(12,27)(13,28)(17,20)(19,23)(21,22)(25,26)(29,30)(31,32);(1,5,6,14,7,21,23,20)(2,10,11,30,12,26,31,28)(3,15,22,19,17,4,16,18)(8,24,29,27,25,32,13,9)Polycyclic presentation:
2^9|a^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319908, #72319954, #72320059, … |
| maximal normal subgroup of | #72319908, #72319954, #72320059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335180);