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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1343 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)(3,6)(4,5)(7,8)(25,26,32,45,31,37,30,36)(27,33,46,55,40,50,39,49)(28,41,42,34,29,35,43,51)(38,48,53,47,44,52,54,56);(2,5)(3,7)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,27)(26,33)(28,38)(29,44)(30,39)(31,40)(32,46)(34,47)(35,52)(36,49)(37,50)(41,48)(42,53)(43,54)(45,55)(51,56);(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,28,31,29)(26,34,37,51)(27,38,40,44)(30,42,32,43)(33,47,50,56)(35,45,41,36)(39,53,46,54)(48,49,52,55);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(16,20)(19,24)(25,29,31,28)(26,35,37,41)(27,38,40,44)(30,42,32,43)(33,48,50,52)(34,45,51,36)(39,54,46,53)(47,49,56,55);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(25,30,31,32)(26,36,37,45)(27,39,40,46)(28,42,29,43)(33,49,50,55)(34,35,51,41)(38,53,44,54)(47,52,56,48)Polycyclic presentation:
2^9|a^2=gi,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,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, 7366947);