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 | |
4183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,11,7)(4,12,14,8)(10,15,13,16)(17,18,23,24)(19,25,20,21)(22,26,30,31)(27,32,28,29)(33,34,38,41)(35,39,44,37)(36,40,46,48)(42,47,43,45);(2,6)(7,9)(8,12)(15,16)(18,24)(21,25)(26,31)(29,32)(33,35,38,44)(34,39,41,37)(36,43,46,42)(40,45,48,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,14)(12,15)(17,20,23,19)(18,25,24,21)(22,27,30,28)(26,29,31,32)(33,36)(34,40)(35,43)(37,47)(38,46)(39,45)(41,48)(42,44);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,21,23,25)(18,20,24,19)(22,29,30,32)(26,28,31,27)(33,35)(34,37)(36,42)(38,44)(39,41)(40,45)(43,46)(47,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10329890);