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 | #42673 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,5)(3,4)(10,12)(14,15)(17,18,23,27)(19,21,28,24)(20,25,31,40)(22,36,37,26)(29,33,32,38)(30,39,42,35)(34,47,44,41)(43,48,45,46);(1,2,6,5)(3,8,4,7)(9,10,13,12)(11,15,16,14)(19,28)(20,31)(21,24)(25,40)(30,42)(34,44)(35,39)(41,47);(1,3,6,4)(2,7,5,8)(9,11,13,16)(10,14,12,15)(17,19)(18,24)(20,32)(21,27)(22,30)(23,28)(25,33)(26,39)(29,31)(34,45)(35,36)(37,42)(38,40)(41,46)(43,44)(47,48);(1,4,6,3)(2,8,5,7)(9,12,13,10)(11,15,16,14)(17,20)(18,25)(19,29)(21,33)(22,34)(23,31)(24,38)(26,41)(27,40)(28,32)(30,43)(35,46)(36,47)(37,44)(39,48)(42,45);(1,5,6,2)(3,8,4,7)(17,21,23,24)(18,19,27,28)(20,33,31,38)(22,35,37,39)(25,29,40,32)(26,30,36,42)(34,46,44,48)(41,43,47,45);(17,22,23,37)(18,26,27,36)(19,30,28,42)(20,34,31,44)(21,35,24,39)(25,41,40,47)(29,43,32,45)(33,46,38,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022637);