Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1300 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,37)(28,32)(29,34)(30,46)(31,36)(33,40)(35,38)(39,53)(41,47)(42,48)(43,49)(44,51)(45,56)(50,55)(52,54);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,38,42,31,40,30,28)(26,32,46,33,36,48,35,37)(29,43,54,41,44,39,45,55)(34,50,56,53,51,47,52,49);(1,3,4,7)(2,6,5,8)(11,17)(14,21)(18,23)(22,24)(25,28,30,40,31,42,38,27)(26,33,35,32,36,37,46,48)(29,41,45,43,44,55,54,39)(34,49,52,47,51,53,56,50);(11,17)(14,21)(18,23)(22,24)(25,29)(26,34)(27,39)(28,41)(30,45)(31,44)(32,47)(33,49)(35,52)(36,51)(37,53)(38,54)(40,43)(42,55)(46,56)(48,50);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(29,44)(34,51)(39,43)(41,55)(45,54)(47,50)(49,53)(52,56)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404627);