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 | |
4162 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)(3,4,8,11)(5,9,13,14)(10,12,15,16)(17,18,22,24)(19,26,30,23)(20,25,27,21)(28,29,31,32)(35,41)(40,47)(42,45)(43,48);(2,7)(4,11)(9,14)(12,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,26)(25,32)(27,31)(33,34,38,39)(35,40,41,47)(36,44,46,37)(42,43,45,48);(5,13)(9,14)(10,15)(12,16)(18,24)(21,25)(23,26)(29,32)(34,39)(37,44)(40,47)(43,48);(1,3)(2,4)(5,10)(6,8)(7,11)(9,12)(13,15)(14,16)(17,20,22,27)(18,25,24,21)(19,28,30,31)(23,32,26,29)(33,35)(34,40)(36,45)(37,43)(38,41)(39,47)(42,46)(44,48);(1,4)(2,8)(3,7)(5,12)(6,11)(9,15)(10,14)(13,16)(17,19)(18,26)(20,31)(21,29)(22,30)(23,24)(25,32)(27,28)(33,36)(34,37)(35,42)(38,46)(39,44)(40,48)(41,45)(43,47);(1,5,6,13)(2,9,7,14)(3,10,8,15)(4,12,11,16)(17,21,22,25)(18,27,24,20)(19,29,30,32)(23,31,26,28)(33,37)(34,36)(35,43)(38,44)(39,46)(40,45)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833489);