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 | #51125 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4002 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)(3,9)(4,6)(5,8)(7,10)(11,15)(12,13)(14,16)(19,25)(23,26)(27,31)(30,32)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,45,48);(2,4)(5,13)(7,9)(11,16)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,35,39,37)(34,40,44,42)(36,38,47,45)(41,43,46,48);(3,10)(7,9)(11,16)(14,15)(18,22)(23,26)(24,28)(30,32)(33,36,39,47)(34,41,44,46)(35,45,37,38)(40,48,42,43);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,37,39,35)(34,42,44,40)(36,38,47,45)(41,43,46,48);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,38)(34,43)(35,36)(37,47)(39,45)(40,41)(42,46)(44,48);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=gh,f^2=gh,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, 8662021);