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 | |
4293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,13,14,9)(11,15,16,12)(17,18)(19,26)(20,21)(22,23)(24,27)(25,31)(28,29)(30,32)(33,34,38,40)(35,39,41,36)(37,45,47,42)(43,44,46,48);(2,7)(4,8)(5,14)(11,16)(18,23)(21,25)(24,26)(29,32)(33,35)(34,39)(36,40)(37,46)(38,41)(42,44)(43,47)(45,48);(2,7)(3,10)(9,13)(11,16)(19,27)(24,26)(28,30)(29,32)(34,40)(35,41)(42,45)(43,46);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,19)(18,24)(20,30)(21,29)(22,27)(23,26)(25,32)(28,31)(33,35)(34,39)(36,40)(37,43)(38,41)(42,48)(44,45)(46,47);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,32)(27,30)(33,36,38,39)(34,41,40,35)(37,44,47,48)(42,46,45,43);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,22,25)(18,20,23,31)(19,29,27,32)(24,30,26,28)(33,37,38,47)(34,42,40,45)(35,43,41,46)(36,44,39,48)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746253);