Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1247 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)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,29,33)(27,35,38,31)(28,40,42,32)(30,34,37,44)(36,47,49,43)(39,45,52,48)(41,53,51,46)(50,54,56,55);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(27,36)(28,41)(30,37)(31,43)(32,46)(34,44)(35,47)(38,49)(39,52)(40,53)(42,51)(45,48);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,37,49,29,38,30,36)(26,31,44,47,33,35,34,43)(28,39,51,56,42,52,41,50)(32,45,53,55,40,48,46,54);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(27,38)(31,35)(36,49)(39,52)(43,47)(45,48)(50,56)(54,55);(11,21)(15,18)(19,20)(23,24)(25,28,30,41,29,42,37,51)(26,32,34,46,33,40,44,53)(27,39,36,50,38,52,49,56)(31,45,43,54,35,48,47,55)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=fgh,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6727687);