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 | |
1273 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)(25,26,30,35,29,34,43,32)(27,36,39,45,38,31,44,47)(28,40,42,48,41,33,49,50)(37,46,53,56,52,51,54,55);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,38)(26,31,34,36)(28,37,41,52)(30,39,43,44)(32,45,35,47)(33,46,40,51)(42,53,49,54)(48,55,50,56);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(30,43)(31,45)(33,48)(34,35)(36,47)(39,44)(40,50)(42,49)(46,55)(51,56)(53,54);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28)(26,33)(27,37)(29,41)(30,42)(31,46)(32,48)(34,40)(35,50)(36,51)(38,52)(39,53)(43,49)(44,54)(45,55)(47,56);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(27,38)(28,41)(31,36)(33,40)(39,44)(42,49)(45,47)(48,50)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fi,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,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, 6775601);