Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,4)(3,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,33,30,36,29,35)(27,32,43,45,39,48,38,47)(28,40,44,50,41,34,42,49)(37,51,54,56,52,46,53,55);(1,2)(3,8)(4,5)(6,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,30,39)(26,32,36,48)(28,37,41,52)(29,38,31,43)(33,45,35,47)(34,46,40,51)(42,53,44,54)(49,55,50,56);(1,3,5,7)(2,6,4,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(29,31)(32,45)(34,49)(35,36)(38,43)(40,50)(42,44)(46,55)(47,48)(51,56)(53,54);(3,7)(6,8)(11,19)(14,17)(18,24)(21,23)(25,28)(26,34)(27,37)(29,42)(30,41)(31,44)(32,46)(33,49)(35,50)(36,40)(38,53)(39,52)(43,54)(45,55)(47,56)(48,51);(1,4,5,2)(3,8,7,6)(10,16)(14,17)(15,22)(21,23)(28,41)(34,40)(37,52)(42,44)(46,51)(49,50)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7394416);