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 | |
4007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,11,12,7)(4,13,14,9)(5,10,15,16)(17,18,22,24)(19,23,26,20)(21,25,30,31)(27,29,32,28);(1,3,6,12)(2,7,8,11)(4,10,14,16)(5,9,15,13)(17,19)(18,23)(20,24)(21,27)(22,26)(25,29)(28,31)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,45)(46,48);(2,8)(4,14)(7,11)(10,16)(18,24)(19,26)(25,31)(27,32)(33,35,38,40)(34,37,41,42)(36,44,46,43)(39,47,48,45);(1,3)(2,7)(4,10)(5,13)(6,12)(8,11)(9,15)(14,16)(17,20)(18,19)(21,29)(22,23)(24,26)(25,32)(27,31)(28,30)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,47)(46,48);(1,4,6,14)(2,9,8,13)(3,10,12,16)(5,7,15,11)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(33,36,38,46)(34,39,41,48)(35,43,40,44)(37,45,42,47);(1,5,6,15)(2,10,8,16)(3,9,12,13)(4,7,14,11)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37)(34,40)(35,41)(36,45)(38,42)(39,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442235);