Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26)(27,32)(28,34)(29,35)(30,45)(31,36)(33,42)(37,52)(38,47)(39,48)(40,55)(41,49)(43,50)(44,51)(46,53)(54,56);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27)(26,32)(28,37)(29,39)(30,40)(31,41)(33,46)(34,52)(35,48)(36,49)(38,43)(42,53)(44,54)(45,55)(47,50)(51,56);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,37)(29,42)(30,44)(31,43)(32,46)(34,45)(35,50)(36,51)(38,41)(39,53)(40,54)(47,48)(49,56)(52,55);(1,3,4,7)(2,6,5,8)(10,17)(15,19)(16,18)(23,24)(25,28)(26,34)(27,38)(29,44)(30,42)(31,43)(32,47)(33,45)(35,51)(36,50)(37,41)(39,53)(40,54)(46,48)(49,52)(55,56);(2,5)(6,8)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(25,29,31,30)(26,35,36,45)(27,39,41,40)(28,42,43,44)(32,48,49,55)(33,50,51,34)(37,53,38,54)(46,47,56,52)Polycyclic presentation:
2^9|c^2=f,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6684193);