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 | |
4113 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)(4,5)(6,8)(9,11)(10,15)(12,13)(14,16)(17,18)(19,20)(21,29)(22,24)(23,27)(25,30)(26,31)(28,32)(35,41)(39,46)(42,44)(47,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,9)(10,14)(15,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,31)(28,29)(30,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(2,8)(3,11)(5,15)(12,14)(18,24)(20,23)(21,30)(26,32)(34,38)(35,41)(40,43)(42,44);(1,3)(2,9)(4,14)(5,13)(6,11)(7,8)(10,12)(15,16)(17,20,22,23)(18,19,24,27)(21,28,30,31)(25,26,29,32)(33,35,37,41)(34,39,38,46)(36,44,45,42)(40,48,43,47);(1,4,6,10)(2,5,8,15)(3,12,11,14)(7,13,9,16)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,31)(30,32)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,5)(2,10)(3,13)(4,8)(6,15)(7,14)(9,12)(11,16)(17,21,22,30)(18,25,24,29)(19,26,27,32)(20,28,23,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)Polycyclic presentation:
2^9|d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837421);