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 | |
4224 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)(19,26)(20,24)(21,23)(22,25)(27,32)(28,30)(29,31)(33,34,38,41)(35,36,43,39)(37,40,47,48)(42,44,45,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,31,30,32)(33,35,38,43)(34,39,41,36)(37,42,47,45)(40,46,48,44);(3,10)(6,7)(11,16)(14,15)(20,28)(24,30)(27,29)(31,32)(33,35,38,43)(34,39,41,36)(37,42,47,45)(40,46,48,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,22,28)(18,24,25,30)(19,27,21,29)(23,31,26,32)(33,35)(34,36)(37,45)(38,43)(39,41)(40,46)(42,47)(44,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,31,30,32)(33,36)(34,35)(37,46)(38,39)(40,45)(41,43)(42,48)(44,47);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,18)(19,23)(20,30)(21,26)(22,25)(24,28)(27,32)(29,31)(33,37)(34,40)(35,42)(36,44)(38,47)(39,46)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=h,b^2=g,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, 8635702);