Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| 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 | #37553 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(25,26,29,33,30,34,43,50)(27,31,37,46,36,45,53,56)(28,38,41,48,40,49,42,32)(35,51,52,47,39,55,54,44);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,24)(18,19)(25,27)(26,31)(28,39)(29,37)(30,36)(32,47)(33,46)(34,45)(35,40)(38,55)(41,54)(42,52)(43,53)(44,48)(49,51)(50,56);(3,8)(5,7)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28)(26,32)(27,35)(29,42)(30,40)(31,44)(33,49)(34,48)(36,39)(37,54)(38,50)(41,43)(45,47)(46,55)(51,56)(52,53);(2,6)(5,7)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(27,36)(28,40)(31,45)(32,48)(37,53)(38,49)(41,42)(46,56);(9,14,15,13)(10,17,18,12)(11,21,22,23)(16,20,19,24)(27,36)(31,45)(35,39)(37,53)(44,47)(46,56)(51,55)(52,54)Polycyclic presentation:
2^9|a^2=ghi,c^2=fhi,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6900646);