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 | |
4300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,9,4,5)(6,10,14,15)(11,16,12,13)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32)(33,34)(35,43)(36,47)(37,48)(38,39)(40,46)(41,45)(42,44);(2,8)(5,9)(10,15)(13,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,34,39,38)(35,40,46,43)(36,41,45,47)(37,48,44,42);(1,3,7,4)(2,9,8,5)(6,11,14,12)(10,16,15,13)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,35)(34,40)(36,44)(37,45)(38,43)(39,46)(41,42)(47,48);(1,4,7,3)(2,9,8,5)(6,11,14,12)(10,13,15,16)(17,18)(19,20)(21,29)(22,25)(23,26)(24,30)(27,32)(28,31)(33,36,39,45)(34,41,38,47)(35,44,46,37)(40,42,43,48);(1,5,7,9)(2,4,8,3)(6,13,14,16)(10,12,15,11)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,37,39,44)(34,42,38,48)(35,45,46,36)(40,41,43,47);(1,6,7,14)(2,10,8,15)(3,11,4,12)(5,13,9,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,34)(35,43,46,40)(36,41,45,47)(37,48,44,42)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315566);