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 | |
4138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,11,15)(4,9,12,16)(5,10,14,13)(19,25)(23,26)(27,31)(30,32)(33,34)(35,40)(36,39)(37,41)(38,46)(42,48)(43,44)(45,47);(2,7)(4,12)(8,15)(10,13)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35)(34,40)(36,43)(37,38)(39,44)(41,46)(42,45)(47,48);(4,12)(5,14)(9,16)(10,13)(18,22)(20,28)(23,26)(27,31)(33,34,39,36)(35,43,44,40)(37,45,47,41)(38,46,48,42);(1,3)(2,8)(4,13)(5,9)(6,11)(7,15)(10,12)(14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,39,34)(35,43,44,40)(37,41,47,45)(38,46,48,42);(1,4,6,12)(2,9,7,16)(3,10,11,13)(5,8,14,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,37)(34,41)(35,38)(36,45)(39,47)(40,46)(42,43)(44,48);(1,5,6,14)(2,10,7,13)(3,9,11,16)(4,8,12,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,38)(34,42)(35,37)(36,46)(39,48)(40,45)(41,43)(44,47)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=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, 8946456);