Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1378 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)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32)(33,34)(35,39)(36,38)(37,44)(40,43)(41,46)(42,48)(45,47);(10,13)(14,16)(18,19)(23,25)(24,26)(31,32)(33,35,38,43)(34,39,36,40)(37,45,46,42)(41,48,44,47);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,48)(44,47)(45,46);(1,3)(2,6)(4,8)(5,7)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36,38,34)(35,40,43,39)(37,44,46,41)(42,48,45,47);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,37)(34,41)(35,42)(36,44)(38,46)(39,47)(40,48)(43,45)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fgi,d^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4207822);