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 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3793 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,23)(20,26)(21,29)(22,24)(25,28)(27,32)(30,31)(33,34,40,39)(35,44,38,41)(36,42,37,43)(45,48,46,47);(3,9)(7,13)(10,15)(14,16)(17,19)(18,23)(20,27)(21,30)(22,28)(24,25)(26,32)(29,31)(33,35,40,38)(34,41,39,44)(36,45,37,46)(42,47,43,48);(2,6)(7,13)(8,11)(14,16)(18,24)(19,28)(26,30)(27,29)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,25)(20,29)(21,30)(22,28)(23,24)(26,32)(27,31)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,26)(19,29)(21,23)(22,31)(24,30)(25,32)(27,28)(33,38,40,35)(34,41,39,44)(36,46,37,45)(42,47,43,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,32)(18,27,24,29)(19,30,28,26)(20,25,31,23)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9622637);