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 | |
4081 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)(19,27)(20,26)(21,24)(22,28)(23,25)(29,32)(30,31)(33,34,38,40)(35,39,43,44)(36,45,46,41)(37,42,48,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,21)(18,24,25,27)(20,31,28,29)(22,30,26,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,44)(45,48)(46,47);(3,10)(6,7)(11,16)(14,15)(18,25)(20,28)(24,27)(29,31)(34,40)(35,43)(41,45)(42,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,26)(19,29)(21,31)(22,25)(23,28)(24,30)(27,32)(33,34)(35,44)(36,41)(37,47)(38,40)(39,43)(42,48)(45,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,23,19)(18,27,25,24)(20,31,28,29)(22,30,26,32)(33,36)(34,41)(35,42)(37,39)(38,46)(40,45)(43,47)(44,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22,23,26)(18,28,25,20)(19,30,21,32)(24,29,27,31)(33,37,38,48)(34,42,40,47)(35,41,43,45)(36,44,46,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,e^2=g,f^2=h,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, 8635911);