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 | #2430 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3971 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34)(35,43)(36,40)(37,41)(38,42)(39,45)(44,48)(46,47);(2,3)(7,9)(8,10)(14,15)(19,28)(20,29)(24,30)(25,32)(33,35,38,45)(34,39,42,43)(36,37,44,47)(40,46,48,41);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,24)(20,21)(22,25)(23,28)(26,29)(27,30)(31,32)(33,36,38,44)(34,40,42,48)(35,37,45,47)(39,46,43,41);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,25)(19,21)(22,24)(23,29)(26,28)(27,32)(30,31)(33,35)(34,39)(36,37)(38,45)(40,46)(41,48)(42,43)(44,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,22)(19,20)(23,26)(24,25)(27,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,45)(44,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22)(18,26)(19,25)(20,30)(21,27)(23,31)(24,29)(28,32)(33,37)(34,41)(35,44)(36,45)(38,47)(39,40)(42,46)(43,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8599758);