Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,8)(10,13)(11,15)(18,22)(19,25)(24,29)(27,31)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(20,28)(24,29)(27,31)(30,32)(34,35)(37,45)(39,41)(43,48);(1,3,5,8)(2,7,6,4)(9,11,12,15)(10,14,13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(1,4,5,7)(2,3,6,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,3)(2,7)(4,6)(5,8)(9,10)(11,16)(12,13)(14,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=f,d^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5429505);