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 | |
4021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,11,16)(4,14,12,6)(5,9,15,13)(17,18)(19,23)(20,24)(21,29)(22,26)(25,31)(27,28)(30,32)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(1,3)(2,8)(4,5)(6,13)(7,11)(9,14)(10,16)(12,15)(17,19)(18,23)(20,26)(21,27)(22,24)(25,32)(28,29)(30,31)(34,40)(35,44)(37,47)(42,45);(3,11)(4,12)(6,14)(8,16)(17,19,22,24)(18,20,26,23)(21,30,31,27)(25,32,29,28)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,4)(2,6)(3,5)(7,12)(8,13)(9,16)(10,14)(11,15)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,13)(14,16)(17,20)(18,24)(19,26)(21,28)(22,23)(25,30)(27,29)(31,32)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,6)(2,4)(3,13)(5,8)(7,14)(9,11)(10,12)(15,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,32)(24,30)(26,29)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)Polycyclic presentation:
2^9|a^2=ghi,c^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, 8835884);