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 | |
4078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,10,5)(4,8,12,16)(11,15,13,14)(17,18)(19,25)(20,28)(21,22)(23,27)(24,30)(26,31)(29,32)(33,34)(35,37)(36,41)(38,48)(39,42)(40,46)(43,45)(44,47);(1,3)(2,7)(4,11)(5,9)(6,10)(8,15)(12,13)(14,16)(18,22)(23,25)(24,28)(31,32)(33,35,39,38)(34,37,42,48)(36,44,46,43)(40,45,41,47);(3,10)(4,12)(5,7)(8,16)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31)(33,36,39,46)(34,40,42,41)(35,43,38,44)(37,47,48,45);(1,3)(2,7)(4,13)(5,9)(6,10)(8,14)(11,12)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,36)(34,41)(35,44)(37,47)(38,43)(39,46)(40,42)(45,48);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,37)(34,35)(36,45)(38,42)(39,48)(40,44)(41,43)(46,47);(1,5,6,7)(2,3,9,10)(4,14,12,15)(8,11,16,13)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,38,39,35)(34,37,42,48)(36,43,46,44)(40,45,41,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8927331);