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 | #52772 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,9,4,6)(5,13,15,10)(11,14,12,16)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28)(33,34,39,42)(35,37,41,38)(36,44,47,40)(43,46,48,45);(2,8)(3,4)(10,13)(11,12)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32)(33,35,39,41)(34,38,42,37)(36,43,47,48)(40,46,44,45);(1,3,7,4)(2,6,8,9)(5,11,15,12)(10,14,13,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,16,13,14)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,36,39,47)(34,40,42,44)(35,43,41,48)(37,45,38,46);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,14,9,16)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,37,39,38)(34,41,42,35)(36,45,47,46)(40,48,44,43);(1,6,7,9)(2,3,8,4)(5,14,15,16)(10,11,13,12)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)(33,38,39,37)(34,35,42,41)(36,46,47,45)(40,43,44,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=i,e^2=i,f^2=g,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, 8021583);