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 | |
3985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,4)(5,13)(6,7)(8,11)(9,15)(10,16)(12,14)(20,28)(23,29)(26,32)(30,31)(33,34,40,37)(35,43,39,41)(36,42,47,38)(44,45,46,48);(2,7)(4,8)(9,13)(12,16)(17,18)(19,24)(20,29)(21,25)(22,27)(23,28)(26,30)(31,32)(33,35,40,39)(34,41,37,43)(36,44,47,46)(38,45,42,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,16)(12,13)(14,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,32)(29,31)(33,36,40,47)(34,42,37,38)(35,44,39,46)(41,48,43,45);(1,4,6,8)(2,3,7,11)(5,12,15,16)(9,10,13,14)(17,20,21,28)(18,23,25,29)(19,26,27,32)(22,30,24,31)(33,37,40,34)(35,41,39,43)(36,38,47,42)(44,48,46,45);(1,3)(2,4)(5,14)(6,11)(7,8)(9,16)(10,15)(12,13)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,32)(29,30)(33,38)(34,36)(35,45)(37,47)(39,48)(40,42)(41,44)(43,46);(1,5,6,15)(2,9,7,13)(3,10,11,14)(4,12,8,16)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)(33,39,40,35)(34,41,37,43)(36,46,47,44)(38,45,42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10388254);