Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,8,4,7)(9,10)(11,16)(12,14)(13,15)(18,22)(19,25)(24,29)(27,31)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(2,6)(3,4)(9,10,13,15)(11,14,12,16)(18,22)(19,25)(20,28)(30,32)(34,38)(35,41)(40,44)(43,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,34)(35,42)(36,44)(37,38)(39,41)(40,45)(43,46)(47,48);(1,4,5,3)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35,37,41)(34,39,38,42)(36,43,45,47)(40,46,44,48);(1,2)(3,8)(4,7)(5,6)(9,10)(11,16)(12,14)(13,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36,37,45)(34,40,38,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fhi,c^2=gi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6176787);