Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1314 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)(10,14)(11,12)(17,18,22,25)(19,26,23,20)(21,29,30,24)(27,28,31,32)(33,34,38,40)(35,41,43,36)(37,39,46,48)(42,47,45,44);(9,10)(11,15)(12,16)(13,14)(19,23)(20,26)(27,31)(28,32)(33,35,38,43)(34,36,40,41)(37,42,46,45)(39,44,48,47);(9,11,13,12)(10,15,14,16)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,35)(34,36)(37,45)(38,43)(39,47)(40,41)(42,46)(44,48);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,14,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,15)(13,14)(17,21,22,30)(18,24,25,29)(19,27,23,31)(20,28,26,32)(33,37)(34,39)(35,42)(36,44)(38,46)(40,48)(41,47)(43,45)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fh,d^2=gh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4669986);