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 | |
3982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,13)(10,15)(17,18,23,25)(19,26,20,22)(21,29,31,24)(27,30,28,32)(34,36)(35,41)(40,44)(43,47);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,18,23,25)(19,26,20,22)(21,24,31,29)(27,32,28,30)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,23,20)(18,22,25,26)(21,27,31,28)(24,30,29,32)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,20,23,19)(18,22,25,26)(21,28,31,27)(24,30,29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,21,23,31)(18,24,25,29)(19,27,20,28)(22,30,26,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,22)(18,19)(20,25)(21,30)(23,26)(24,27)(28,29)(31,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gi,d^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8103577);