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 | |
4103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,12)(3,13,6,8)(4,10,15,9)(5,14,16,11)(20,28)(23,31)(27,32)(29,30)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3,7,6)(2,8,12,13)(4,14,15,11)(5,9,16,10)(17,18)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32)(33,35,39,37)(34,40,38,42)(36,43,46,44)(41,47,45,48);(1,4)(2,9)(3,14)(5,13)(6,11)(7,15)(8,16)(10,12)(17,19)(18,21)(20,27)(22,25)(23,30)(24,26)(28,32)(29,31)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4)(2,10)(3,11)(5,13)(6,14)(7,15)(8,16)(9,12)(17,20,22,28)(18,23,26,31)(19,27,25,32)(21,30,24,29)(33,35,39,37)(34,42,38,40)(36,43,46,44)(41,48,45,47);(1,5,7,16)(2,11,12,14)(3,9,6,10)(4,13,15,8)(17,19)(18,24)(20,27)(21,26)(22,25)(23,29)(28,32)(30,31)(33,37,39,35)(34,40,38,42)(36,44,46,43)(41,47,45,48);(1,6,7,3)(2,8,12,13)(4,11,15,14)(5,9,16,10)(17,21,22,24)(18,25,26,19)(20,29,28,30)(23,27,31,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8595543);