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 | |
4046 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18)(19,21)(20,25)(22,26)(23,24)(27,29)(28,30)(31,32)(33,34)(35,40)(36,44)(37,38)(39,43)(41,47)(42,48)(45,46);(3,9)(6,13)(10,15)(14,16)(18,24)(21,25)(26,32)(29,30)(33,35)(34,40)(36,37)(38,44)(39,45)(41,48)(42,47)(43,46);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19,23,20)(18,21,24,25)(22,27,31,28)(26,29,32,30)(33,36,39,47)(34,41,43,44)(35,42,45,37)(38,46,48,40);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,30,32,29)(33,36)(34,41)(35,37)(38,46)(39,47)(40,48)(42,45)(43,44);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,21,23,25)(18,19,24,20)(22,30,31,29)(26,28,32,27)(33,37)(34,38)(35,36)(39,42)(40,44)(41,46)(43,48)(45,47);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,22,23,31)(18,26,24,32)(19,27,20,28)(21,29,25,30)(33,38)(34,42)(35,44)(36,46)(37,43)(39,48)(40,47)(41,45)Polycyclic presentation:
2^9|c^2=gh,d^2=g,e^2=g,f^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, 9817313);