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 | |
4072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,11)(8,14)(10,15)(13,16)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34,39,43)(35,38,45,47)(36,42,46,40)(37,41,44,48);(1,3,5,11)(2,6,7,9)(4,13,14,10)(8,16,12,15)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,39,45)(34,38,43,47)(36,37,46,44)(40,48,42,41);(2,7)(6,9)(8,12)(15,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,36,39,46)(34,40,43,42)(35,44,45,37)(38,41,47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,15)(12,16)(13,14)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,37)(34,41)(35,36)(38,42)(39,44)(40,47)(43,48)(45,46);(1,4,5,14)(2,8,7,12)(3,10,11,13)(6,15,9,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38)(34,35)(36,41)(37,42)(39,47)(40,44)(43,45)(46,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,36)(34,42)(35,37)(38,41)(39,46)(40,43)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=i,d^2=ghi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8176922);