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 | #40649 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,10)(9,15)(11,13)(33,34,39,45)(35,40,47,48)(36,42,46,41)(37,43,44,38);(1,2,6,7)(3,8,10,5)(4,9,12,15)(11,16,13,14)(17,18,22,23)(19,24,27,32)(20,25,30,29)(21,26,31,28)(33,35,39,47)(34,40,45,48)(36,44,46,37)(38,41,43,42);(3,10)(4,12)(5,8)(9,15)(18,23)(19,27)(20,30)(21,31)(33,36,39,46)(34,41,45,42)(35,37,47,44)(38,48,43,40);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,12)(15,16)(17,19)(18,24)(20,26)(21,29)(22,27)(23,32)(25,31)(28,30)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,45)(43,48);(1,4,6,12)(2,9,7,15)(3,11,10,13)(5,14,8,16)(17,20)(18,25)(19,28)(21,24)(22,30)(23,29)(26,27)(31,32)(33,37,39,44)(34,43,45,38)(35,46,47,36)(40,41,48,42);(1,5,6,8)(2,3,7,10)(4,14,12,16)(9,11,15,13)(17,21)(18,26)(19,29)(20,24)(22,31)(23,28)(25,27)(30,32)(33,38,39,43)(34,44,45,37)(35,41,47,42)(36,40,46,48)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8886594);