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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18,22,24)(19,26,30,23)(20,25,27,21)(28,29,31,32)(33,34,38,40)(35,39,41,36)(37,42,46,47)(43,45,48,44);(2,7)(4,8)(10,14)(12,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,26)(25,32)(27,31)(33,35,38,41)(34,39,40,36)(37,43,46,48)(42,45,47,44);(5,13)(10,14)(11,16)(12,15)(18,24)(21,25)(23,26)(29,32)(34,40)(35,41)(42,47)(43,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,20,22,27)(18,25,24,21)(19,28,30,31)(23,32,26,29)(33,36,38,39)(34,41,40,35)(37,45,46,44)(42,43,47,48);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,19)(18,26)(20,31)(21,29)(22,30)(23,24)(25,32)(27,28)(33,37,38,46)(34,42,40,47)(35,43,41,48)(36,44,39,45);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,21,22,25)(18,27,24,20)(19,29,30,32)(23,31,26,28)(33,34)(35,39)(36,41)(37,42)(38,40)(43,45)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833595);