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 | |
3902 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)(3,11)(4,6)(5,8)(7,9)(10,16)(12,15)(13,14)(19,25)(22,29)(26,31)(30,32)(33,34)(35,37)(36,40)(38,41)(39,42)(43,45)(44,46)(47,48);(1,3,7,5)(2,8,9,11)(4,12,10,14)(6,13,16,15)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,46,48,45);(2,9)(4,10)(8,11)(12,14)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36,39,35)(34,40,42,37)(38,44,47,43)(41,46,48,45);(1,4,7,10)(2,6,9,16)(3,12,5,14)(8,13,11,15)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31);(1,5,7,3)(2,8,9,11)(4,14,10,12)(6,13,16,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,37,39,40)(34,35,42,36)(38,46,47,45)(41,44,48,43);(1,6)(2,10)(3,13)(4,9)(5,15)(7,16)(8,14)(11,12)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=h,e^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, 9820530);