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 | |
4000 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)(3,8)(4,6)(5,14)(7,9)(10,13)(11,16)(12,15)(19,25)(22,26)(27,31)(30,32)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(1,3)(2,8)(4,13)(5,11)(6,10)(7,12)(9,15)(14,16)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(34,39)(35,42)(41,44)(43,48);(2,9)(3,12)(4,11)(10,14)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4,7,11)(2,6,9,16)(3,13,12,5)(8,10,15,14)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,5,7,13)(2,10,9,14)(3,11,12,4)(6,15,16,8)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,6)(2,11)(3,14)(4,9)(5,15)(7,16)(8,13)(10,12)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|c^2=ghi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664214);