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 | |
4153 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)(3,5,10,8)(4,9,12,15)(11,14,13,16)(17,18)(19,27)(20,31)(21,26)(22,23)(24,30)(25,29)(28,32);(2,7)(3,10)(4,12)(14,16)(17,19,23,30)(18,24,22,27)(20,21,29,28)(25,26,31,32)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,47)(46,48);(3,10)(4,12)(5,8)(9,15)(18,22)(20,29)(21,28)(24,27)(34,38)(35,42)(36,45)(47,48);(1,3,6,10)(2,8,7,5)(4,13,12,11)(9,14,15,16)(17,20)(18,25)(19,28)(21,30)(22,31)(23,29)(24,32)(26,27)(33,35)(34,39)(36,46)(37,42)(38,41)(40,48)(43,45)(44,47);(1,4,6,12)(2,9,7,15)(3,11,10,13)(5,14,8,16)(17,21,23,28)(18,26,22,32)(19,29,30,20)(24,31,27,25)(33,36)(34,40)(35,43)(37,45)(38,44)(39,47)(41,48)(42,46);(1,5,6,8)(2,10,7,3)(4,14,12,16)(9,13,15,11)(17,22,23,18)(19,27,30,24)(20,31,29,25)(21,32,28,26)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965719);