Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,33)(28,42)(29,35)(30,36)(31,46)(32,37)(34,43)(38,53)(39,48)(40,49)(41,51)(44,52)(45,50)(47,55)(54,56);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27)(26,33)(28,38)(29,39)(30,45)(31,40)(32,41)(34,47)(35,48)(36,50)(37,51)(42,53)(43,55)(44,54)(46,49)(52,56);(11,17)(14,21)(18,23)(22,24)(25,28,32,29)(26,34,37,52)(27,38,41,39)(30,43,31,44)(33,47,51,56)(35,46,42,36)(40,54,45,55)(48,49,53,50);(1,3,4,7)(2,6,5,8)(11,17)(14,21)(18,23)(22,24)(25,29,32,28)(26,35,37,42)(27,39,41,38)(30,43,31,44)(33,48,51,53)(34,46,52,36)(40,54,45,55)(47,49,56,50);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,30,32,31)(26,36,37,46)(27,40,41,45)(28,43,29,44)(33,49,51,50)(34,35,52,42)(38,54,39,55)(47,53,56,48)Polycyclic presentation:
2^9|a^2=fh,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6404278);