Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(11,15)(14,16)(17,18,21,22)(19,23,27,25)(20,24,29,28)(26,30,32,31)(33,34)(35,37)(36,43)(38,40)(39,41)(42,45)(44,47)(46,48);(2,5)(6,8)(10,13)(14,16)(18,22)(23,25)(24,28)(30,31)(33,35,39,36)(34,37,41,43)(38,44,46,42)(40,47,48,45);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,36,39,35)(34,37,41,43)(38,44,46,42)(40,45,48,47);(9,11)(10,14)(12,15)(13,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,37)(34,36)(35,41)(38,45)(39,43)(40,44)(42,48)(46,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,38,39,46)(34,40,41,48)(35,42,36,44)(37,45,43,47)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fhi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5400450);