Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300980 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ 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:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,36)(27,32,39,48)(28,34,42,52)(29,35,30,33)(37,46,40,50)(38,45,43,47)(41,49,44,51)(53,56,54,55);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,32)(28,40)(29,43)(30,38)(31,39)(33,45)(34,50)(35,47)(36,48)(37,42)(41,53)(44,54)(46,52)(49,56)(51,55);(1,3)(2,5)(4,8)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(29,30)(32,45)(34,49)(35,36)(38,43)(41,44)(46,55)(47,48)(50,56)(51,52)(53,54);(2,6)(3,8)(11,19)(14,17)(18,24)(21,23)(25,28)(26,34)(27,37)(29,44)(30,41)(31,42)(32,46)(33,49)(35,51)(36,52)(38,54)(39,40)(43,53)(45,55)(47,56)(48,50);(2,6)(5,7)(10,16)(14,17)(15,22)(21,23)(25,29,31,30)(26,35,36,33)(27,38,39,43)(28,41,42,44)(32,45,48,47)(34,49,52,51)(37,53,40,54)(46,56,50,55)Polycyclic presentation:
2^9|a^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7209377);