Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(10,13)(11,15)(17,18)(19,25)(20,28)(21,22)(23,27)(24,30)(26,31)(29,32)(33,34,37,38)(35,41,43,39)(36,40,45,44)(42,47,48,46);(1,2,5,4)(3,6,7,8)(11,15)(14,16)(18,22)(23,25)(24,28)(31,32)(34,38)(39,41)(40,44)(46,47);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,34)(35,39)(36,44)(37,38)(40,45)(41,43)(42,47)(46,48);(1,4,5,2)(3,8,7,6)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fh,b^2=ghi,c^2=ghi,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4835925);