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 | |
4334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,8,11,7)(5,9,13,12)(10,15,16,14)(17,18)(19,28)(20,26)(21,27)(22,29)(23,25)(24,30)(31,32)(35,41)(39,46)(42,44)(47,48);(1,3,6,11)(2,7,4,8)(5,10,13,16)(9,14,12,15)(17,19,23,30)(18,24,25,28)(20,21,29,31)(22,27,26,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(5,13)(9,12)(10,16)(14,15)(18,25)(22,26)(24,28)(27,32)(34,38)(39,46)(40,43)(47,48);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,20)(18,26)(19,21)(22,25)(23,29)(24,32)(27,28)(30,31)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,21)(18,27)(19,29)(20,30)(22,28)(23,31)(24,26)(25,32)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,22,23,26)(18,20,25,29)(19,27,30,32)(21,28,31,24)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832432);