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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1294 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,35,29,34,44,32)(27,36,39,46,38,31,45,48)(28,40,43,49,42,33,50,52)(37,47,53,56,41,51,54,55);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,31)(28,41)(29,38)(30,39)(32,46)(33,51)(34,36)(35,48)(37,42)(40,47)(43,54)(44,45)(49,56)(50,53)(52,55);(2,5)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(30,44)(31,46)(33,49)(34,35)(36,48)(39,45)(40,52)(43,50)(47,55)(51,56)(53,54);(10,16)(14,17)(15,22)(21,23)(25,28)(26,33)(27,37)(29,42)(30,43)(31,47)(32,49)(34,40)(35,52)(36,51)(38,41)(39,53)(44,50)(45,54)(46,55)(48,56);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(21,23)(27,38)(28,42)(31,36)(33,40)(39,45)(43,50)(46,48)(49,52)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6709032);