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 | #2543 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,10,4)(5,13,15,9)(11,12,16,14)(17,18,23,28)(19,22,27,24)(20,25,30,32)(21,26,29,31)(34,39)(35,37)(41,44)(43,45);(2,7)(4,8)(9,13)(12,14)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,26)(31,32)(33,34,38,39)(35,40,37,42)(36,44,46,41)(43,48,45,47);(2,7)(3,10)(9,13)(11,16)(17,20,23,30)(18,25,28,32)(19,29,27,21)(22,31,24,26)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,26)(28,32)(33,35,38,37)(34,40,39,42)(36,43,46,45)(41,47,44,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,21,23,29)(18,26,28,31)(19,30,27,20)(22,32,24,25)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,22)(18,27)(19,28)(20,31)(21,25)(23,24)(26,30)(29,32)(33,37,38,35)(34,42,39,40)(36,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113610);