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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1380 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)(10,13)(11,15)(17,18)(19,22)(20,25)(21,26)(23,27)(24,30)(28,31)(29,32)(33,34,37,40)(35,41,42,38)(36,39,44,46)(43,47,48,45);(1,2,5,4)(3,6,7,8)(10,13)(14,16)(18,19)(20,27)(24,26)(28,29)(35,42)(38,41)(43,48)(45,47);(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,27,23)(21,26,30,24)(28,32,29,31)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,35,37,42)(34,38,40,41)(36,43,44,48)(39,45,46,47);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,21,22,30)(18,24,19,26)(20,28,27,29)(23,31,25,32)(33,36,37,44)(34,39,40,46)(35,43,42,48)(38,45,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=f,d^2=gh,e^2=i,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, 4825835);