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 | |
4201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,5,10)(6,9,14,12)(11,15,13,16)(17,18)(19,23)(20,25)(21,31)(22,27)(24,30)(26,28)(29,32)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,16)(13,14)(17,19,22,29)(18,23,27,32)(20,28,30,21)(24,31,25,26)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,2,7,4)(3,10,5,8)(6,9,14,12)(11,16,13,15)(17,20)(18,24)(19,28)(21,29)(22,30)(23,31)(25,27)(26,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,4,7,2)(3,10,5,8)(6,9,14,12)(11,15,13,16)(17,20)(18,25)(19,28)(21,29)(22,30)(23,26)(24,27)(31,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,18)(19,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21,22,28)(18,26,27,31)(19,20,29,30)(23,24,32,25)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gi,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10352124);