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 | |
4187 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)(3,10)(4,12)(5,15)(6,8)(7,13)(9,14)(11,16)(17,18,23,25)(19,28,29,24)(20,30,27,22)(21,26,31,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,10)(9,16)(14,15)(17,19)(18,24)(20,21)(22,26)(23,29)(25,28)(27,31)(30,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,8)(4,14)(7,10)(11,15)(18,25)(19,29)(20,27)(26,32)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,4,6,14)(2,9,8,12)(3,11,13,15)(5,7,16,10)(17,20,23,27)(18,22,25,30)(19,21,29,31)(24,26,28,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,10)(4,11)(5,9)(6,13)(7,8)(12,16)(14,15)(17,21)(18,26)(19,27)(20,29)(22,28)(23,31)(24,30)(25,32)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5)(2,11)(3,12)(4,7)(6,16)(8,15)(9,13)(10,14)(17,22)(18,27)(19,26)(20,25)(21,28)(23,30)(24,31)(29,32)(33,36)(34,39)(35,41)(37,44)(38,45)(40,43)(42,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9690137);