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 | |
4401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,11,9)(4,12,14,7)(10,16,13,15)(17,18,22,25)(19,26,20,23)(21,29,30,24)(27,31,28,32)(33,34)(35,43)(36,46)(37,42)(38,39)(40,45)(41,47)(44,48);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,35)(34,40)(36,44)(37,47)(38,43)(39,45)(41,48)(42,46);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36,39,47)(34,41,38,46)(35,44,45,37)(40,48,43,42);(1,2)(3,9)(4,7)(5,8)(6,11)(10,16)(12,14)(13,15)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,37)(34,42)(35,36)(38,48)(39,44)(40,41)(43,46)(45,47);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,35)(34,43)(36,44)(37,47)(38,40)(39,45)(41,42)(46,48);(1,4)(2,7)(3,10)(5,14)(6,15)(8,12)(9,16)(11,13)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,38,39,34)(35,43,45,40)(36,46,47,41)(37,48,44,42)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10411784);