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 | |
4230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18)(19,24)(20,22)(21,28)(23,26)(25,29)(27,30)(31,32)(34,37)(35,36)(42,45)(43,44);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(19,26)(23,24)(27,32)(30,31)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(2,6)(7,9)(8,13)(14,15)(17,19,22,26)(18,23,20,24)(21,27,29,32)(25,30,28,31)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,20,22,18)(19,24,26,23)(21,28,29,25)(27,31,32,30)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21,22,29)(18,25,20,28)(19,27,26,32)(23,30,24,31)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081804);