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 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(11,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(34,35)(36,43)(37,45)(46,47);(10,13)(14,16)(18,22)(20,27)(23,29)(25,31)(33,34,38,35)(36,39,43,41)(37,40,45,42)(44,46,48,47);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,28)(21,22)(24,27)(25,32)(26,29)(30,31)(33,35,38,34)(36,39,43,41)(37,40,45,42)(44,47,48,46);(1,2)(3,5)(4,6)(7,8)(17,19,21,26)(18,23,22,29)(20,25,27,31)(24,30,28,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,3,4,7)(2,5,6,8)(9,11,12,15)(10,14,13,16)(17,20,21,27)(18,24,22,28)(19,25,26,31)(23,30,29,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=fg,e^2=hi,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, 5497035);