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 | |
4348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,7,13,16)(10,15,12,14)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,46)(47,48);(1,3,5,11)(2,6,8,9)(4,10,13,12)(7,14,16,15)(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(33,36,38,42)(34,39,41,35)(37,44,46,47)(40,45,48,43);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,35)(37,44)(38,42)(39,41)(40,43)(45,48)(46,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,37,38,46)(34,40,41,48)(35,43,39,45)(36,44,42,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)(33,35)(34,36)(37,45)(38,39)(40,47)(41,42)(43,46)(44,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817389);