Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | |
1310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,8)(9,10,13,17)(11,15,18,14)(12,20,22,16)(19,23,21,24)(25,26,30,34)(27,35,39,49)(28,32,40,46)(29,42,36,33)(31,43,45,38)(37,48,51,56)(41,53,50,47)(44,54,55,52);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,14,17,15)(12,21,22,19)(16,24,20,23);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,24)(20,23)(21,22)(25,27,36,43,30,39,29,38)(26,31,42,49,34,45,33,35)(28,37,50,54,40,51,41,52)(32,44,53,56,46,55,47,48);(2,5)(6,8)(11,18)(14,15)(19,21)(23,24)(25,28)(26,32)(27,37)(29,41)(30,40)(31,44)(33,47)(34,46)(35,48)(36,50)(38,52)(39,51)(42,53)(43,54)(45,55)(49,56);(9,12,13,22)(10,16,17,20)(11,19,18,21)(14,23,15,24)(28,40)(32,46)(37,51)(41,50)(44,55)(47,53)(48,56)(52,54)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=fghi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105280);