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 | |
4192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18)(19,21)(20,24)(22,25)(23,26)(27,29)(28,30)(31,32)(33,34,38,39)(35,37,41,46)(36,40,44,47)(42,45,48,43);(4,11)(8,14)(10,15)(12,16)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,30,32,29)(33,35)(34,37)(36,42)(38,41)(39,46)(40,45)(43,47)(44,48);(2,7)(3,9)(8,14)(10,15)(34,39)(35,41)(40,47)(42,48);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,20,23,19)(18,24,26,21)(22,28,31,27)(25,30,32,29)(33,34)(35,37)(36,40)(38,39)(41,46)(42,45)(43,48)(44,47);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21,23,24)(18,19,26,20)(22,30,31,29)(25,28,32,27)(33,36,38,44)(34,40,39,47)(35,42,41,48)(37,45,46,43);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,22,23,31)(18,25,26,32)(19,27,20,28)(21,29,24,30)(33,37)(34,35)(36,43)(38,46)(39,41)(40,48)(42,47)(44,45)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,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^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9818000);