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 | |
4310 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,20)(21,29)(22,24)(23,27)(25,30)(26,31)(28,32)(33,34,39,42)(35,40,43,48)(36,46,44,38)(37,41,47,45);(2,6)(7,10)(8,13)(14,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,31)(28,29)(30,32)(33,35)(34,40)(36,37)(38,45)(39,43)(41,46)(42,48)(44,47);(3,9)(4,12)(7,10)(8,13)(18,24)(20,23)(21,30)(26,32)(35,43)(36,44)(38,46)(40,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,20,22,23)(18,19,24,27)(21,28,30,31)(25,26,29,32)(33,36)(34,38)(35,37)(39,44)(40,45)(41,48)(42,46)(43,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,31)(30,32)(33,37)(34,41)(35,44)(36,43)(38,40)(39,47)(42,45)(46,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,30)(18,25,24,29)(19,26,27,32)(20,28,23,31)(33,38)(34,36)(35,45)(37,40)(39,46)(41,43)(42,44)(47,48)Polycyclic presentation:
2^9|a^2=i,d^2=h,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, 8835969);