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 | |
4293 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)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18)(19,27)(20,23)(21,25)(22,31)(24,29)(26,32)(28,30)(33,34,39,44)(35,40,46,37)(36,42,48,41)(38,47,45,43);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(33,35,39,46)(34,40,44,37)(36,45,48,38)(41,47,42,43);(2,7)(3,10)(8,13)(11,16)(18,20)(24,27)(25,30)(26,31)(33,36,39,48)(34,41,44,42)(35,38,46,45)(37,47,40,43);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20,23,18)(19,27,29,24)(21,30,28,25)(22,31,32,26)(33,36)(34,42)(35,45)(37,47)(38,46)(39,48)(40,43)(41,44);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21)(18,25)(19,22)(20,30)(23,28)(24,26)(27,31)(29,32)(33,37)(34,35)(36,47)(38,41)(39,40)(42,45)(43,48)(44,46);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,15)(10,16)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,38,39,45)(34,43,44,47)(35,36,46,48)(37,42,40,41)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9722523);