Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1217 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26)(27,31)(28,33)(29,32)(30,34)(35,44)(36,43)(37,45)(38,42)(39,46)(40,48)(41,47)(49,54)(50,53)(51,55)(52,56);(1,3,5,4)(2,6,7,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,29,37)(26,31,32,45)(28,36,39,51)(30,35,41,38)(33,43,46,55)(34,44,47,42)(40,49,52,50)(48,54,56,53);(1,4,5,3)(2,6,7,8)(26,32)(27,35)(28,39)(30,41)(31,42)(36,50)(37,38)(43,54)(44,45)(48,56)(49,51)(53,55);(10,15)(14,17)(16,23)(22,24)(25,28,29,39)(26,33,32,46)(27,36,37,51)(30,40,41,52)(31,43,45,55)(34,48,47,56)(35,49,38,50)(42,53,44,54);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,35)(30,41)(31,44)(34,47)(36,49)(37,38)(40,52)(42,45)(43,54)(48,56)(50,51)(53,55)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6601141);