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 | |
4115 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)(3,4,11,7)(5,8,13,15)(10,12,16,14)(17,18,23,24)(19,27,30,22)(20,31,29,25)(21,32,28,26);(1,3,6,11)(2,7,9,4)(5,10,13,16)(8,14,15,12)(17,19,23,30)(18,22,24,27)(20,28,29,21)(25,26,31,32)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,48,46,43);(5,13)(8,15)(10,16)(12,14)(18,24)(22,27)(25,31)(26,32)(34,39)(37,45)(40,41)(43,48);(1,4,6,7)(2,3,9,11)(5,12,13,14)(8,10,15,16)(17,20)(18,25)(19,28)(21,30)(22,26)(23,29)(24,31)(27,32)(33,35,38,44)(34,40,39,41)(36,46,47,42)(37,43,45,48);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,21,23,28)(18,26,24,32)(19,29,30,20)(22,25,27,31)(33,36)(34,37)(35,42)(38,47)(39,45)(40,48)(41,43)(44,46);(1,5,6,13)(2,8,9,15)(3,10,11,16)(4,12,7,14)(17,22,23,27)(18,19,24,30)(20,26,29,32)(21,25,28,31)(33,37)(34,36)(35,43)(38,45)(39,47)(40,46)(41,42)(44,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300844);