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 | |
4134 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)(3,11)(4,10)(5,8)(6,13)(7,9)(12,16)(14,15)(18,23)(20,24)(25,30)(28,31)(35,41)(36,44)(39,42)(40,45);(1,3,7,5)(2,8,9,11)(4,13,14,12)(6,10,16,15)(17,18,22,23)(19,20,27,24)(21,25,29,30)(26,28,32,31)(33,34)(35,42)(36,45)(37,38)(39,41)(40,44)(43,47)(46,48);(2,9)(6,16)(8,11)(10,15)(21,29)(25,30)(26,32)(28,31)(34,38)(39,42)(40,45)(47,48);(1,4)(2,10)(3,12)(5,13)(6,8)(7,14)(9,15)(11,16)(17,19,22,27)(18,24,23,20)(21,26,29,32)(25,31,30,28)(33,35)(34,39)(36,46)(37,41)(38,42)(40,48)(43,44)(45,47);(1,5,7,3)(2,11,9,8)(4,12,14,13)(6,15,16,10)(17,20,22,24)(18,19,23,27)(21,28,29,31)(25,26,30,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,47)(41,46)(42,48);(1,6)(2,12)(3,10)(4,8)(5,15)(7,16)(9,13)(11,14)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48)Polycyclic presentation:
2^9|b^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839971);