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 | |
4223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,8,4,7)(5,9,13,16)(11,15,12,14)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32)(33,34,39,43)(35,38,37,40)(36,42,47,41)(44,48,45,46);(1,3,6,4)(2,7,10,8)(5,11,13,12)(9,14,16,15)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,35,39,37)(34,40,43,38)(36,45,47,44)(41,46,42,48);(1,3,6,4)(2,8,10,7)(5,11,13,12)(9,15,16,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,36)(34,41)(35,44)(37,45)(38,46)(39,47)(40,48)(42,43);(1,4,6,3)(2,8,10,7)(5,11,13,12)(9,14,16,15)(17,18)(19,20)(21,29)(22,25)(23,26)(24,30)(27,32)(28,31)(33,37,39,35)(34,38,43,40)(36,45,47,44)(41,46,42,48);(1,2)(3,8)(4,7)(5,9)(6,10)(11,15)(12,14)(13,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,36)(34,42)(35,45)(37,44)(38,46)(39,47)(40,48)(41,43);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38)(34,35)(36,46)(37,43)(39,40)(41,44)(42,45)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=g,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, 10315330);