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 | |
4003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34,39,43)(35,44,41,37)(36,40,38,42)(45,46,48,47);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(33,35)(34,37)(36,45)(38,48)(39,41)(40,47)(42,46)(43,44);(2,6)(7,10)(8,12)(14,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,36,39,38)(34,40,43,42)(35,45,41,48)(37,47,44,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,37,39,44)(34,41,43,35)(36,46,38,47)(40,45,42,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,46)(47,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38,39,36)(34,42,43,40)(35,45,41,48)(37,47,44,46)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,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=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662018);