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 | |
3900 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,18,22,28)(19,26,30,23)(20,25,31,24)(21,29,32,27)(35,41)(39,45)(42,47)(46,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19)(18,23)(20,29)(21,25)(22,30)(24,32)(26,28)(27,31)(34,38)(39,45)(40,43)(46,48);(17,20,22,31)(18,24,28,25)(19,29,30,27)(21,23,32,26)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20)(18,25)(19,29)(21,23)(22,31)(24,28)(26,32)(27,30)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,2)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,19)(18,26)(20,29)(21,24)(22,30)(23,28)(25,32)(27,31)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,21,22,32)(18,27,28,29)(19,25,30,24)(20,23,31,26)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^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, 9817889);