Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #34633 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,16)(13,15)(21,33)(26,35)(29,40)(31,44)(37,45)(38,43)(42,47)(46,48);(2,3)(6,7)(9,11,13,16)(10,14,15,12)(18,23)(19,27)(25,30)(26,35)(28,39)(29,40)(38,43)(42,47);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,18)(19,24)(20,30)(21,26)(22,23)(25,32)(27,34)(28,41)(29,37)(31,43)(33,35)(36,39)(38,44)(40,45)(42,48)(46,47);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,19,22,27)(18,24,23,34)(20,28,32,39)(21,29,33,40)(25,36,30,41)(26,37,35,45)(31,42,44,47)(38,46,43,48);(17,20,22,32)(18,25,23,30)(19,28,27,39)(21,31,33,44)(24,36,34,41)(26,38,35,43)(29,42,40,47)(37,46,45,48);(17,21)(18,26)(19,29)(20,31)(22,33)(23,35)(24,37)(25,38)(27,40)(28,42)(30,43)(32,44)(34,45)(36,46)(39,47)(41,48)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551496);