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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1377 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)(25,26,29,34)(27,36,38,31)(28,33,41,49)(30,32,43,35)(37,48,40,45)(39,46,52,44)(42,47,54,50)(51,56,53,55);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27,29,38)(26,31,34,36)(28,40,41,37)(30,39,43,52)(32,44,35,46)(33,48,49,45)(42,53,54,51)(47,56,50,55);(11,17)(14,20)(18,23)(21,24)(26,32)(30,43)(31,44)(33,47)(34,35)(36,46)(39,52)(42,54)(45,55)(48,56)(49,50)(51,53);(1,3,4,7)(2,6,5,8)(10,15)(14,20)(16,22)(21,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,45)(32,47)(34,49)(35,50)(36,48)(38,40)(39,51)(43,54)(44,55)(46,56)(52,53);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(28,41)(33,49)(37,40)(42,54)(45,48)(47,50)(51,53)(55,56)Polycyclic presentation:
2^9|a^2=i,b^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,d^b=di,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, 6344321);