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) = C25×C4 |
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:
(2,5)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18,22,25)(19,20,26,23)(21,29,30,24)(27,31,32,28)(33,34)(35,39)(36,42)(37,40)(38,41)(43,46)(44,48)(45,47);(9,11)(10,14)(12,15)(13,16)(19,26)(20,23)(27,32)(28,31)(33,35,38,36)(34,39,41,42)(37,43,45,44)(40,46,47,48);(10,15)(11,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,31)(28,29)(30,32)(33,36,38,35)(34,39,41,42)(37,44,45,43)(40,46,47,48);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20,22,23)(18,19,25,26)(21,28,30,31)(24,27,29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,3,4,7)(2,6,5,8)(9,11)(10,12)(13,16)(14,15)(17,21,22,30)(18,24,25,29)(19,27,26,32)(20,28,23,31)(33,37,38,45)(34,40,41,47)(35,43,36,44)(39,46,42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=gh,d^2=fh,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, 4669861);