Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | |
1607 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)(17,18,22,27)(19,28,31,38)(20,33,23,25)(21,26,37,42)(24,29,40,32)(30,43,47,48)(34,41,36,35)(39,44,45,46);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,18)(19,28)(20,25)(21,35)(22,27)(23,33)(24,29)(26,36)(30,45)(31,38)(32,40)(34,42)(37,41)(39,47)(43,44)(46,48);(17,19,22,31)(18,24,27,40)(20,29,23,32)(21,30,37,47)(25,38,33,28)(26,39,42,45)(34,44,36,46)(35,48,41,43);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,20,22,23)(18,25,27,33)(19,29,31,32)(21,36,37,34)(24,38,40,28)(26,41,42,35)(30,46,47,44)(39,43,45,48);(3,11)(6,9)(10,16)(14,15)(17,21,22,37)(18,26,27,42)(19,30,31,47)(20,34,23,36)(24,39,40,45)(25,35,33,41)(28,43,38,48)(29,44,32,46)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579837);