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 | |
4137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,14,16)(17,18,23,26)(19,20,21,24)(22,30,31,25)(27,32,29,28)(33,34)(35,41)(36,40)(37,39)(38,42)(43,48)(44,47)(45,46);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,14)(15,16)(17,19)(18,20)(21,23)(22,27)(24,26)(25,28)(29,31)(30,32)(33,35,38,37)(34,39,42,41)(36,44,46,43)(40,48,45,47);(3,11)(7,8)(12,14)(13,16)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,34,38,42)(35,39,37,41)(36,40,46,45)(43,47,44,48);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,30)(31,32)(33,34)(35,39)(36,45)(37,41)(38,42)(40,46)(43,48)(44,47);(1,4)(2,5)(3,12)(6,9)(7,13)(8,16)(10,15)(11,14)(17,21,23,19)(18,24,26,20)(22,27,31,29)(25,28,30,32)(33,36)(34,40)(35,43)(37,44)(38,46)(39,47)(41,48)(42,45);(1,5)(2,9)(3,13)(4,10)(6,15)(7,14)(8,12)(11,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,37,38,35)(34,41,42,39)(36,43,46,44)(40,47,45,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^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=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, 9078814);