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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1418 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,32,37)(27,33,40,51)(28,41,29,35)(30,36,31,46)(34,44,52,42)(38,48,43,53)(39,49,45,50)(47,54,56,55);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,33)(28,38)(29,43)(30,45)(31,39)(32,40)(34,47)(35,53)(36,50)(37,51)(41,48)(42,55)(44,54)(46,49)(52,56);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,32,29)(26,34,37,52)(27,38,40,43)(30,42,31,44)(33,47,51,56)(35,46,41,36)(39,54,45,55)(48,50,53,49);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,29,32,28)(26,35,37,41)(27,38,40,43)(30,42,31,44)(33,48,51,53)(34,46,52,36)(39,55,45,54)(47,50,56,49);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(25,30,32,31)(26,36,37,46)(27,39,40,45)(28,42,29,44)(33,49,51,50)(34,35,52,41)(38,54,43,55)(47,48,56,53)Polycyclic presentation:
2^9|a^2=i,c^2=fi,d^2=i,e^2=i,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,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, 7108033);