Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10,14,15)(11,16,13,12)(18,22)(20,28)(23,26)(27,31)(33,34)(35,40)(36,38)(37,44)(39,43)(41,46)(42,47)(45,48);(3,8)(5,7)(10,15)(11,13)(18,22)(19,25)(24,29)(27,31)(33,35,38,43)(34,39,36,40)(37,45,46,42)(41,48,44,47);(9,11,14,13)(10,16,15,12)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,36,38,34)(35,39,43,40)(37,44,46,41)(42,47,45,48);(1,3)(2,5)(4,8)(6,7)(9,12,14,16)(10,11,15,13)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,48)(44,47)(45,46);(1,2)(3,5)(4,6)(7,8)(9,13,14,11)(10,16,15,12)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37)(34,41)(35,42)(36,44)(38,46)(39,47)(40,48)(43,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=fgh,c^2=h,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5909153);