Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1334 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)(3,5)(4,7)(6,8)(10,13)(11,15)(18,22)(19,25)(20,28)(30,32)(33,34,38,36)(35,40,43,39)(37,44,46,41)(42,47,45,48);(1,3)(2,5)(4,8)(6,7)(10,13)(14,16)(33,35,38,43)(34,39,36,40)(37,45,46,42)(41,48,44,47);(1,2,4,7)(3,6,8,5)(9,10)(11,16)(12,13)(14,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,48)(44,47)(45,46);(1,2)(3,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,34)(35,40,43,39)(37,44,46,41)(42,48,45,47);(1,3)(2,6)(4,8)(5,7)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,46)(34,41,36,44)(35,42,43,45)(39,47,40,48)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=ghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6018142);