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 | |
4049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,18,22,25)(19,21,24,26)(20,23,31,29)(27,30,28,32)(33,34)(35,45)(36,38)(37,42)(39,43)(40,44)(41,48)(46,47);(2,3)(4,11)(8,10)(12,16)(17,19)(18,21)(20,28)(22,24)(23,32)(25,26)(27,31)(29,30)(33,35,40,48)(34,41,44,45)(36,46,37,43)(38,39,42,47);(5,13)(8,10)(12,16)(14,15)(19,24)(21,26)(27,28)(30,32)(33,36,40,37)(34,42,44,38)(35,46,48,43)(39,41,47,45);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,18)(19,26)(20,29)(21,24)(22,25)(23,31)(27,30)(28,32)(33,37,40,36)(34,42,44,38)(35,43,48,46)(39,41,47,45);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,23)(19,27)(21,30)(22,31)(24,28)(25,29)(26,32)(33,38,40,42)(34,37,44,36)(35,47,48,39)(41,46,45,43);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,26)(18,24,25,19)(20,30,31,32)(23,28,29,27)(33,39,40,47)(34,43,44,46)(35,42,48,38)(36,45,37,41)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325660);