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 | |
4025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,28)(31,32)(34,37)(35,41)(40,44)(43,47);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32)(33,34,38,37)(35,42,41,39)(36,44,45,40)(43,46,47,48);(2,6)(7,14)(8,11)(15,16)(17,18,22,26)(19,20,24,23)(21,29,30,25)(27,31,32,28)(33,35,38,41)(34,39,37,42)(36,43,45,47)(40,46,44,48);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32)(33,34,38,37)(35,39,41,42)(36,40,45,44)(43,46,47,48);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,20)(18,24)(19,26)(21,28)(22,23)(25,32)(27,29)(30,31)(33,36,38,45)(34,40,37,44)(35,43,41,47)(39,46,42,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37,38,34)(35,42,41,39)(36,44,45,40)(43,48,47,46)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,e^2=g,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, 8832201);