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 | |
4240 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,24)(22,25)(23,27)(26,29)(28,32)(30,31)(33,34,38,42)(35,41,46,39)(36,37,47,40)(43,48,44,45);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,30)(29,32)(33,35)(34,39)(36,43)(37,45)(38,46)(40,48)(41,42)(44,47);(3,10)(6,7)(11,16)(14,15)(20,28)(24,32)(26,30)(29,31)(33,36,38,47)(34,40,42,37)(35,43,46,44)(39,48,41,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,22,28)(18,24,25,32)(19,26,27,30)(21,29,23,31)(33,36)(34,37)(35,44)(38,47)(39,48)(40,42)(41,45)(43,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,22,23)(18,19,25,27)(20,29,28,31)(24,26,32,30)(33,37)(34,36)(35,45)(38,40)(39,43)(41,44)(42,47)(46,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,19)(18,21)(20,30)(22,27)(23,25)(24,31)(26,28)(29,32)(33,35)(34,41)(36,43)(37,48)(38,46)(39,42)(40,45)(44,47)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635700);