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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,8)(5,6)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,32)(28,39)(29,44)(30,35)(31,36)(33,42)(34,45)(37,52)(38,47)(40,50)(41,49)(43,55)(46,54)(48,56)(51,53);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,30,38)(26,32,35,47)(28,40,42,56)(29,31,45,41)(33,48,39,50)(34,49,44,36)(37,53,54,43)(46,55,52,51);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,41,53,30,42,31,43)(26,33,49,55,35,39,36,51)(27,37,45,56,38,54,29,40)(32,46,34,50,47,52,44,48);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(28,42)(33,39)(37,54)(40,56)(43,53)(46,52)(48,50)(51,55);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,31,38,30,45,41,27)(26,34,36,32,35,44,49,47)(28,40,43,54,42,56,53,37)(33,50,51,46,39,48,55,52)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fhi,d^2=fgi,e^2=fghi,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, 6753920);