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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(6,8)(9,10)(11,15)(12,13)(14,16)(19,27)(23,28)(29,32)(30,41)(37,39)(38,42)(43,45)(47,48);(3,7)(6,8)(10,13)(14,15)(17,18,22,26)(19,23,27,28)(20,31,34,24)(21,25,36,35)(29,39,32,37)(30,38,41,42)(33,44,46,40)(43,48,45,47);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,28)(20,24)(21,35)(22,26)(23,27)(25,36)(29,39)(30,38)(31,34)(32,37)(33,44)(40,46)(41,42)(43,47)(45,48);(9,11)(10,14)(12,16)(13,15)(17,19,22,27)(18,23,26,28)(20,32,34,29)(21,30,36,41)(24,39,31,37)(25,38,35,42)(33,45,46,43)(40,48,44,47);(1,3)(2,6)(4,8)(5,7)(17,20,22,34)(18,24,26,31)(19,29,27,32)(21,33,36,46)(23,37,28,39)(25,40,35,44)(30,43,41,45)(38,47,42,48);(1,4,5,2)(3,8,7,6)(17,21)(18,25)(19,30)(20,33)(22,36)(23,38)(24,40)(26,35)(27,41)(28,42)(29,43)(31,44)(32,45)(34,46)(37,47)(39,48)Polycyclic presentation:
2^9|b^2=gi,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839742);