Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
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:
(3,7)(6,8)(10,13)(14,16)(19,25)(20,27)(23,28)(24,29)(33,34,38,41)(35,42,36,39)(37,40,45,48)(43,47,44,46);(11,15)(14,16)(18,22)(23,28)(24,29)(30,32)(33,35,38,36)(34,39,41,42)(37,44,45,43)(40,47,48,46);(1,2,5,4)(3,6,7,8)(9,10,12,13)(11,14,15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,36,38,35)(34,39,41,42)(37,44,45,43)(40,46,48,47);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,34)(35,39)(36,42)(37,40)(38,41)(43,46)(44,47)(45,48);(1,4,5,2)(3,8,7,6)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,45)(34,40,41,48)(35,43,36,44)(39,46,42,47)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4496991);