Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1363 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)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,32,37)(27,33,41,50)(28,35,29,44)(30,36,31,46)(34,45,52,43)(38,53,42,48)(39,49,40,56)(47,55,51,54);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27)(26,33)(28,42)(29,38)(30,39)(31,40)(32,41)(34,51)(35,48)(36,49)(37,50)(43,55)(44,53)(45,54)(46,56)(47,52);(11,17)(14,20)(18,23)(21,24)(25,28,32,29)(26,34,37,52)(27,38,41,42)(30,43,31,45)(33,47,50,51)(35,36,44,46)(39,54,40,55)(48,56,53,49);(1,3,4,7)(2,6,5,8)(10,15)(14,20)(16,22)(21,24)(25,29,32,28)(26,35,37,44)(27,38,41,42)(30,43,31,45)(33,48,50,53)(34,36,52,46)(39,55,40,54)(47,56,51,49);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(25,30,32,31)(26,36,37,46)(27,39,41,40)(28,43,29,45)(33,49,50,56)(34,44,52,35)(38,54,42,55)(47,48,51,53)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344340);