Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1336 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)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26)(27,35)(28,39)(29,31)(30,43)(32,38)(33,40)(34,42)(36,48)(37,51)(41,47)(44,45)(46,56)(49,50)(52,53)(54,55);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,31)(27,36)(33,39)(34,43)(37,50)(38,44)(47,54)(52,56);(3,7)(5,8)(25,27,29,36)(26,32,31,45)(28,37,40,50)(30,44,42,38)(33,46,39,53)(34,48,43,35)(41,56,55,52)(47,51,54,49);(2,6)(3,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,33)(27,37)(29,40)(30,41)(31,39)(32,46)(34,47)(35,49)(36,50)(38,52)(42,55)(43,54)(44,56)(45,53)(48,51);(3,7)(5,8)(10,16)(14,17)(15,24)(22,23)(26,31)(28,40)(32,45)(34,43)(35,48)(37,50)(41,55)(52,56)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,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, 7110408);