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 | |
3844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,22,25)(19,20,27,23)(21,29,31,24)(26,32,30,28)(33,34,39,44)(35,43,38,40)(36,46,45,41)(37,42,47,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19)(18,20)(21,30)(22,27)(23,25)(24,32)(26,31)(28,29)(33,35,39,38)(34,40,44,43)(36,47,45,37)(41,48,46,42);(2,6)(7,10)(8,12)(14,16)(17,19,22,27)(18,23,25,20)(21,26,31,30)(24,32,29,28)(33,36)(34,41)(35,37)(38,47)(39,45)(40,42)(43,48)(44,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,37,39,47)(34,42,44,48)(35,45,38,36)(40,46,43,41);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,31)(18,24,25,29)(19,26,27,30)(20,28,23,32)(33,38,39,35)(34,43,44,40)(36,47,45,37)(41,48,46,42)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663321);