Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| 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 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(3,4)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,31,36)(27,32,39,49)(28,34,29,42)(30,44,45,35)(33,43,51,41)(37,52,40,47)(38,54,56,48)(46,55,50,53);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27)(26,32)(28,40)(29,37)(30,38)(31,39)(33,50)(34,47)(35,48)(36,49)(41,55)(42,52)(43,53)(44,54)(45,56)(46,51);(2,6)(7,8)(25,28,31,29)(26,33,36,51)(27,37,39,40)(30,43,45,41)(32,46,49,50)(34,35,42,44)(38,55,56,53)(47,54,52,48);(1,3,5,4)(2,7,6,8)(11,18)(14,17)(19,24)(22,23)(25,29,31,28)(26,34,36,42)(27,37,39,40)(30,43,45,41)(32,47,49,52)(33,35,51,44)(38,53,56,55)(46,54,50,48);(1,4,5,3)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(28,29)(33,51)(34,42)(37,40)(41,43)(46,50)(47,52)(53,55)Polycyclic presentation:
2^9|a^2=fh,c^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,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, 7134809);