Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1254 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,35)(28,33)(29,42)(30,32)(31,38)(34,43)(36,50)(37,49)(39,54)(40,46)(41,56)(44,53)(45,51)(47,55)(48,52);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,29,38,30,36,43,51)(26,31,34,35,32,45,42,50)(28,39,41,55,40,37,52,53)(33,47,48,54,46,44,56,49);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(26,32)(27,36)(33,46)(34,42)(37,39)(38,51)(48,56)(53,55);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,28,30,40)(26,33,32,46)(27,37,36,39)(29,41,43,52)(31,44,45,47)(34,48,42,56)(35,49,50,54)(38,53,51,55);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,32)(27,36)(28,40)(34,42)(38,51)(41,52)(44,47)(49,54)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,d^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405478);