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 | |
1263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,13,7,14)(3,11,16,23,17,24,5,9)(4,10,19,27,20,28,21,29)(15,30,31,22,32,26,18,25);(1,3)(2,9)(4,15)(5,6)(7,16)(8,17)(10,25)(11,14)(12,24)(13,23)(18,19)(20,32)(21,31)(22,28)(26,27)(29,30);(1,4)(2,10)(3,15)(5,18)(6,19)(7,21)(8,20)(9,22)(11,26)(12,27)(13,28)(14,29)(16,31)(17,32)(23,25)(24,30);(1,5,6,17,8,16,7,3)(2,11,12,9,13,24,14,23)(4,18,19,32,20,31,21,15)(10,26,27,22,28,30,29,25)Polycyclic presentation:
2^9|a^2=e,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318890, #72318940, #72318969, … |
| maximal normal subgroup of | #72318890, #72318940, #72318969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335179);