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 | |
4100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,10,4,5)(6,8,15,16)(11,14,12,13)(17,18)(19,24)(20,29)(21,26)(22,30)(23,25)(27,31)(28,32)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(1,3,7,4)(2,5,9,10)(6,11,15,12)(8,13,16,14)(17,19,23,27)(18,24,25,31)(20,28,30,21)(22,26,29,32)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,46)(47,48);(18,25)(19,27)(20,30)(26,32)(33,36,38,42)(34,39,41,35)(37,44,46,47)(40,45,48,43);(1,4,7,3)(2,5,9,10)(6,12,15,11)(8,13,16,14)(17,20)(18,22)(19,28)(21,27)(23,30)(24,26)(25,29)(31,32)(33,36)(34,35)(37,44)(38,42)(39,41)(40,43)(45,48)(46,47);(1,5)(2,3)(4,9)(6,14)(7,10)(8,12)(11,16)(13,15)(17,21,23,28)(18,26,25,32)(19,20,27,30)(22,24,29,31)(33,37,38,46)(34,40,41,48)(35,43,39,45)(36,44,42,47);(1,6)(2,8)(3,11)(4,12)(5,13)(7,15)(9,16)(10,14)(17,22)(18,20)(19,26)(21,31)(23,29)(24,28)(25,30)(27,32)(33,35)(34,36)(37,45)(38,39)(40,47)(41,42)(43,46)(44,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=hi,d^2=ghi,e^2=h,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, 9830858);