Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,23)(21,24)(25,26,29,34,30,35,41,32)(27,31,37,44,38,45,51,42)(28,33,40,48,39,47,52,46)(36,43,50,55,49,54,56,53);(2,5)(3,7)(9,11,13,18)(10,14,17,15)(12,20,21,19)(16,23,24,22)(25,27,30,38)(26,31,35,45)(28,36,39,49)(29,37,41,51)(32,42,34,44)(33,43,47,54)(40,50,52,56)(46,53,48,55);(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24)(26,32)(29,41)(31,42)(33,46)(34,35)(37,51)(40,52)(43,53)(44,45)(47,48)(50,56)(54,55);(1,3,4,7)(2,6,5,8)(11,18)(14,15)(19,20)(22,23)(25,28,30,39)(26,33,35,47)(27,36,38,49)(29,40,41,52)(31,43,45,54)(32,46,34,48)(37,50,51,56)(42,53,44,55);(2,5)(6,8)(9,12,13,21)(10,16,17,24)(11,19,18,20)(14,22,15,23)(28,39)(33,47)(36,49)(40,52)(43,54)(46,48)(50,56)(53,55)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7394680);