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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3923 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)(3,8)(4,6)(5,10)(7,9)(11,13)(12,15)(14,16)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34,39,41)(35,36,44,40)(37,42,48,46)(38,45,47,43);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,15,16,13)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35,39,44)(34,40,41,36)(37,47,48,38)(42,45,46,43);(2,9)(3,4)(10,16)(11,12)(18,20)(19,26)(25,28)(27,32)(34,41)(36,40)(42,46)(43,45);(1,4,7,3)(2,6,9,8)(5,11,14,12)(10,13,16,15)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36)(34,35)(37,43)(38,46)(39,40)(41,44)(42,47)(45,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,15,8,13)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,37,39,48)(34,42,41,46)(35,38,44,47)(36,45,40,43);(1,6)(2,3)(4,9)(5,13)(7,8)(10,12)(11,16)(14,15)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,38)(34,43)(35,37)(36,46)(39,47)(40,42)(41,45)(44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10124242);