Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682648 |
| 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 | #61224 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1621 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,28)(20,33)(21,26)(22,25)(23,27)(24,30)(29,45)(31,34)(32,43)(35,39)(36,48)(37,41)(38,44)(40,46)(42,47);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,24)(20,34)(21,36)(22,30)(23,31)(25,28)(26,42)(27,43)(29,38)(32,33)(35,45)(37,47)(39,46)(40,44)(41,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,33,18,22,27,23,25)(19,24,32,43,30,28,31,34)(21,35,39,26,37,44,38,41)(29,40,36,48,46,45,47,42);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(18,25)(20,27)(21,37)(24,28)(29,46)(34,43)(36,47)(38,39);(3,13)(7,10)(11,12)(15,16)(17,21,22,37)(18,26,25,41)(19,29,30,46)(20,35,27,44)(23,38,33,39)(24,40,28,45)(31,47,32,36)(34,42,43,48)Polycyclic presentation:
2^9|c^2=fg,d^2=fi,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,d^c=di,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, 6591769);