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 | |
4273 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)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34,38,39)(35,41,46,40)(36,42,47,45)(37,44,48,43);(4,11)(8,14)(10,15)(12,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(34,39)(36,47)(40,41)(43,44);(2,7)(3,9)(8,14)(10,15)(18,22)(23,25)(24,28)(30,31)(33,35,38,46)(34,40,39,41)(36,44,47,43)(37,45,48,42);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35)(34,41)(36,44)(37,45)(38,46)(39,40)(42,48)(43,47);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,36)(34,42)(35,44)(37,40)(38,47)(39,45)(41,48)(43,46);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37,38,48)(34,43,39,44)(35,45,46,42)(36,41,47,40)Polycyclic presentation:
2^9|a^2=h,c^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9627681);