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 | |
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)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,24,26,20)(21,25,29,31)(27,32,30,28)(33,34,38,41)(35,39,36,43)(37,40,45,48)(42,46,44,47);(11,15)(13,16)(18,23)(19,26)(21,29)(28,32)(33,35,38,36)(34,39,41,43)(37,44,45,42)(40,47,48,46);(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,23)(21,30)(22,26)(25,28)(27,29)(31,32)(33,36,38,35)(34,39,41,43)(37,44,45,42)(40,46,48,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,24)(18,19,23,26)(21,28,29,32)(25,27,31,30)(33,34)(35,39)(36,43)(37,40)(38,41)(42,46)(44,47)(45,48);(1,3,4,7)(2,6,5,8)(9,10)(11,13)(12,14)(15,16)(17,21,22,29)(18,25,23,31)(19,27,26,30)(20,28,24,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,48)(43,47)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=f,e^2=ghi,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, 5465547);