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 | |
4294 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)(3,4)(5,13)(6,8)(7,11)(9,14)(10,15)(12,16)(17,18)(19,22)(20,29)(21,25)(23,26)(24,28)(27,32)(30,31)(33,34)(35,39)(36,40)(37,45)(38,42)(41,47)(43,44)(46,48);(1,3)(2,7)(4,8)(5,10)(6,11)(9,15)(12,13)(14,16)(17,19)(18,22)(20,21)(23,27)(24,30)(25,29)(26,32)(28,31)(33,35)(34,39)(36,42)(37,43)(38,40)(41,48)(44,45)(46,47);(2,8)(4,7)(5,14)(10,16)(19,27)(20,28)(22,32)(24,29)(33,35,38,40)(34,36,42,39)(37,46,47,43)(41,48,45,44);(1,4,6,7)(2,3,8,11)(5,12,14,15)(9,10,13,16)(17,20)(18,24)(19,21)(22,30)(23,28)(25,32)(26,29)(27,31)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,15)(14,16)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,30)(29,32)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,13)(11,16)(17,22)(18,19)(20,30)(21,24)(23,32)(25,28)(26,27)(29,31)(33,37)(34,41)(35,43)(36,44)(38,47)(39,48)(40,46)(42,45)Polycyclic presentation:
2^9|c^2=gh,d^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, 8834181);