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 | |
4103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(10,14)(11,16)(12,15)(17,18,23,21)(19,26,20,24)(22,29,30,25)(27,31,28,32)(34,38)(35,36)(42,45)(43,44);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18,23,21)(19,24,20,26)(22,25,30,29)(27,31,28,32)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(2,7)(4,8)(10,14)(12,15)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,35,39,36)(34,40,38,41)(37,44,46,43)(42,48,45,47);(1,3,6,9)(2,8,7,4)(5,11,13,16)(10,15,14,12)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,36,39,35)(34,41,38,40)(37,44,46,43)(42,48,45,47);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,21,23,18)(19,24,20,26)(22,29,30,25)(27,31,28,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002872);