Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1300 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)(3,4)(5,7)(6,8)(9,10,13,11)(12,15,16,14)(18,22)(19,25)(20,28)(30,32)(33,34)(35,36)(37,41)(38,39)(40,42)(43,44)(45,47)(46,48);(1,3,6,5)(2,7,8,4)(18,22)(20,28)(23,26)(27,31)(34,39)(35,42)(37,45)(44,46);(1,4,6,7)(2,3,8,5)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35)(34,40)(36,39)(37,43)(38,42)(41,46)(44,47)(45,48);(1,4)(2,3)(5,8)(6,7)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,40)(34,35,39,42)(37,46,45,44)(41,48,47,43);(1,5,6,3)(2,7,8,4)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,45)(34,41,39,47)(35,43,42,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=fhi,b^2=gh,c^2=ghi,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5638012);