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 | |
4162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,8,11)(5,13,15,9)(10,14,16,12)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,34,38,40)(35,36,42,44)(37,39,48,47)(41,43,45,46);(2,7)(4,11)(9,13)(12,14)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35)(34,36)(37,45)(38,42)(39,46)(40,44)(41,48)(43,47);(2,7)(3,8)(9,13)(10,16)(19,26)(22,23)(27,32)(30,31)(33,36,38,44)(34,35,40,42)(37,43,48,46)(39,41,47,45);(1,3)(2,4)(5,10)(6,8)(7,11)(9,12)(13,14)(15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36)(34,35)(37,46)(38,44)(39,45)(40,42)(41,47)(43,48);(1,4)(2,8)(3,7)(5,14)(6,11)(9,10)(12,15)(13,16)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,37)(34,39)(35,41)(36,43)(38,48)(40,47)(42,45)(44,46);(1,5,6,15)(2,9,7,13)(3,10,8,16)(4,12,11,14)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,34)(35,36)(37,47)(38,40)(39,48)(41,46)(42,44)(43,45)Polycyclic presentation:
2^9|a^2=h,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9073985);