Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424610 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2477 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15390 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,4,8,11)(3,13,15,25,16,9,23,27)(5,18,22,10,20,26,21,17)(14,29,31,24,19,32,30,28);(1,3)(2,9)(4,13)(5,19)(6,23)(7,16)(8,15)(10,24)(11,27)(12,25)(14,20)(17,28)(18,32)(21,30)(22,31)(26,29);(1,4)(2,7)(3,9)(5,17)(6,11)(8,12)(10,20)(13,16)(14,24)(15,27)(18,22)(19,28)(21,26)(23,25)(29,30)(31,32);(1,5,6,22,7,20,8,21)(2,10,12,26,4,17,11,18)(3,14,15,31,16,19,23,30)(9,24,27,32,13,28,25,29)Polycyclic presentation:
2^9|a^2=ef,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336001);