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 | |
4217 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)(19,25)(23,30)(26,28)(31,32)(34,38)(36,44)(39,41)(42,48);(2,7)(4,8)(10,14)(12,15)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(5,13)(10,14)(11,16)(12,15)(18,22)(23,30)(24,27)(31,32)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|c^2=h,d^2=hi,e^2=g,f^2=h,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, 9388635);