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 | |
4061 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,34)(35,37)(36,39)(38,44)(40,42)(41,46)(43,47)(45,48);(2,7)(4,8)(10,14)(12,15)(20,27)(24,30)(26,31)(28,32)(33,35)(34,40)(36,37)(38,43)(39,42)(41,47)(44,45)(46,48);(5,13)(10,14)(11,16)(12,15)(18,23)(19,25)(24,30)(26,31)(34,36)(35,42)(41,44)(43,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,21)(18,19)(20,28)(22,29)(23,25)(24,26)(27,32)(30,31)(33,38,39,46)(34,41,36,44)(35,43,42,48)(37,45,40,47)Polycyclic presentation:
2^9|a^2=hi,d^2=i,e^2=gh,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^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388696);