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 | |
1310 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)(3,8)(4,7)(5,6)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(24,29)(27,31)(33,34,38,36)(35,39,41,43)(37,44,46,40)(42,48,45,47);(2,5)(3,4)(9,11,12,15)(10,13,14,16)(18,22)(19,25)(20,28)(30,32)(34,36)(35,41)(40,44)(42,45);(1,3,6,4)(2,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,39)(36,43)(37,45)(38,41)(40,48)(42,46)(44,47);(1,4,6,3)(2,7,5,8)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,34)(35,39,41,43)(37,44,46,40)(42,47,45,48);(1,5,6,2)(3,7,4,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=gi,d^2=f,e^2=g,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, 6176778);