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) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,13,7)(4,10,14,16)(5,12,15,11)(19,25)(23,30)(26,28)(31,32)(33,34)(35,40)(36,44)(37,42)(38,45)(39,41)(43,47)(46,48);(1,3)(2,7)(4,12)(5,10)(6,13)(8,9)(11,14)(15,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35)(34,40)(36,41)(37,38)(39,44)(42,45)(43,48)(46,47);(2,8)(5,15)(7,9)(10,16)(18,22)(23,30)(24,27)(31,32)(34,41)(35,44)(38,47)(42,48);(1,3)(2,9)(4,11)(5,10)(6,13)(7,8)(12,14)(15,16)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,36,39,40)(34,35,41,44)(37,45,46,43)(38,42,47,48);(1,4)(2,10)(3,12)(5,7)(6,14)(8,16)(9,15)(11,13)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37)(34,42)(35,38)(36,43)(39,46)(40,45)(41,48)(44,47);(1,5,6,15)(2,11,8,12)(3,10,13,16)(4,7,14,9)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,38)(34,43)(35,37)(36,42)(39,47)(40,48)(41,45)(44,46)Polycyclic presentation:
2^9|a^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832377);