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 | |
4073 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)(3,6)(4,11)(5,8)(7,12)(9,15)(10,14)(13,16)(19,28)(21,31)(23,24)(26,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,45)(44,48)(46,47);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,18,22,27)(19,24,28,23)(20,25,30,32)(21,26,31,29)(34,39)(36,40)(42,47)(44,45);(1,2,5,8)(3,6,10,14)(4,7,12,11)(9,13,16,15)(17,19,22,28)(18,23,27,24)(20,29,30,26)(21,25,31,32)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,45,47,44);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,19)(18,24)(20,29)(21,25)(22,28)(23,27)(26,30)(31,32);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20)(18,25)(19,29)(21,24)(22,30)(23,31)(26,28)(27,32)(33,36,38,40)(34,41,39,35)(37,45,46,44)(42,43,47,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,21)(18,26)(19,25)(20,23)(22,31)(24,30)(27,29)(28,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,47)(40,45)(41,48)Polycyclic presentation:
2^9|b^2=h,c^2=gh,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, 9817390);