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 | |
1420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,12,13)(11,14,15,16)(19,25)(20,27)(23,28)(24,29)(33,34)(35,43)(36,38)(37,40)(39,42)(41,48)(44,45)(46,47);(11,15)(14,16)(18,22)(23,28)(24,29)(30,32)(33,35,39,37)(34,40,42,43)(36,44,41,46)(38,47,48,45);(10,13)(14,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,36)(34,38)(35,44)(37,46)(39,41)(40,47)(42,48)(43,45);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,37,39,35)(34,40,42,43)(36,46,41,44)(38,47,48,45);(1,3,4,7)(2,5,6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,38)(34,41)(35,45)(36,42)(37,47)(39,48)(40,44)(43,46)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=h,d^2=fh,e^2=hi,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, 5509869);