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 | |
4088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,12,8,4)(5,9,13,14)(6,15,16,10)(17,18,23,27)(19,22,21,24)(20,26,31,25)(28,32,29,30)(33,34,40,43)(35,38,42,48)(36,45,47,41)(37,39,46,44);(1,3,7,8)(2,4,11,12)(5,6,13,16)(9,10,14,15)(17,19,23,21)(18,24,27,22)(20,29,31,28)(25,30,26,32)(33,35,40,42)(34,38,43,48)(36,44,47,39)(37,41,46,45);(1,4)(2,8)(3,11)(5,10)(6,14)(7,12)(9,16)(13,15)(17,20)(18,25)(19,28)(21,29)(22,30)(23,31)(24,32)(26,27)(33,36,40,47)(34,41,43,45)(35,44,42,39)(37,38,46,48);(1,4)(2,3)(5,15)(6,14)(7,12)(8,11)(9,16)(10,13)(17,21,23,19)(18,22,27,24)(20,29,31,28)(25,30,26,32)(33,37)(34,39)(35,41)(36,38)(40,46)(42,45)(43,44)(47,48);(1,5)(2,9)(3,6)(4,10)(7,13)(8,16)(11,14)(12,15)(17,20)(18,26)(19,29)(21,28)(22,30)(23,31)(24,32)(25,27)(33,38)(34,42)(35,43)(36,46)(37,47)(39,41)(40,48)(44,45);(1,6)(2,10)(3,13)(4,14)(5,8)(7,16)(9,12)(11,15)(17,22)(18,19)(20,30)(21,27)(23,24)(25,28)(26,29)(31,32)(33,39)(34,37)(35,36)(38,41)(40,44)(42,47)(43,46)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10388223);