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 | |
3909 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,30)(31,32)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(3,9)(7,13)(10,15)(14,16)(17,18,23,28)(19,24,20,25)(21,22,27,26)(29,32,31,30)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(2,6)(7,13)(8,11)(14,16)(17,19,23,20)(18,24,28,25)(21,29,27,31)(22,32,26,30)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,25,28,24)(21,29,27,31)(22,32,26,30)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,27)(18,26,28,22)(19,29,20,31)(24,30,25,32)(33,35,39,38)(34,42,36,40)(37,43,46,45)(41,48,44,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,26)(18,27,28,21)(19,30,20,32)(24,29,25,31)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388680);