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 | |
4089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,7)(3,5,9,4)(6,14,15,10)(11,12,16,13)(20,28)(23,31)(27,32)(29,30)(33,34,40,39)(35,45,47,41)(36,42,46,48)(37,43,38,44);(1,3)(2,4)(5,7)(6,11)(8,9)(10,12)(13,14)(15,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32)(33,35)(34,41)(36,37)(38,46)(39,45)(40,47)(42,44)(43,48);(1,4,8,5)(2,9,7,3)(6,12,15,13)(10,16,14,11)(17,19)(18,21)(20,27)(22,25)(23,30)(24,26)(28,32)(29,31)(33,36,40,46)(34,42,39,48)(35,37,47,38)(41,44,45,43);(1,5,8,4)(2,9,7,3)(6,13,15,12)(10,16,14,11)(17,20,22,28)(18,23,26,31)(19,27,25,32)(21,30,24,29)(33,37,40,38)(34,43,39,44)(35,36,47,46)(41,48,45,42);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,19)(18,24)(20,27)(21,26)(22,25)(23,29)(28,32)(30,31)(33,38,40,37)(34,44,39,43)(35,46,47,36)(41,42,45,48);(1,7,8,2)(3,5,9,4)(6,10,15,14)(11,12,16,13)(17,21,22,24)(18,25,26,19)(20,29,28,30)(23,27,31,32)(33,39,40,34)(35,45,47,41)(36,48,46,42)(37,43,38,44)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=gi,e^2=gh,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, 8595585);