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 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10,14,12)(11,15,13,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34)(35,39)(36,43)(37,38)(40,44)(41,47)(42,45)(46,48);(9,11,14,13)(10,15,12,16)(20,28)(23,30)(26,32)(29,31)(34,37)(39,45)(40,43)(46,47);(9,12,14,10)(11,15,13,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,35)(34,39)(36,41)(37,45)(38,42)(40,46)(43,47)(44,48);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,16,12,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,36,38,44)(34,40,37,43)(35,41,42,48)(39,46,45,47);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,15,12,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,37,38,34)(35,39,42,45)(36,43,44,40)(41,46,48,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=fh,e^2=gi,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, 5463435);