Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3013 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18,22,26)(19,28,32,23)(20,34,37,42)(21,27,39,25)(24,36,43,38)(29,46,47,41)(30,31,33,44)(35,40,48,45);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19,22,32)(18,23,26,28)(20,35,37,48)(21,30,39,33)(24,41,43,46)(25,31,27,44)(29,36,47,38)(34,45,42,40);(17,20,22,37)(18,24,26,43)(19,29,32,47)(21,36,39,38)(23,40,28,45)(25,42,27,34)(30,35,33,48)(31,46,44,41);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,22,39)(18,25,26,27)(19,30,32,33)(20,36,37,38)(23,31,28,44)(24,42,43,34)(29,35,47,48)(40,46,45,41);(3,11)(6,9)(10,16)(14,15)(17,18)(19,31)(20,24)(21,25)(22,26)(23,33)(27,39)(28,30)(29,46)(32,44)(34,38)(35,45)(36,42)(37,43)(40,48)(41,47)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579496);