Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(11,15)(12,16)(19,25)(20,27)(23,28)(24,29)(33,34)(35,38)(36,39)(37,42)(40,46)(41,43)(44,48)(45,47);(2,5)(6,8)(9,10,13,14)(11,16,15,12)(18,22)(23,28)(24,29)(30,32)(34,35)(36,43)(40,42)(44,45);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,34)(36,41,43,39)(37,42,46,40)(44,48,45,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,43)(34,39,35,41)(37,45,46,44)(40,48,42,47);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|b^2=fhi,c^2=fgh,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4783858);