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 | |
3919 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,34,39,40)(35,37,42,38)(36,44,47,41)(43,46,48,45);(3,9)(7,13)(10,15)(14,16)(18,23)(20,24)(26,30)(28,31)(34,40)(35,42)(41,44)(43,48);(2,6)(7,13)(8,11)(14,16)(21,29)(26,30)(27,32)(28,31)(33,35)(34,37)(36,43)(38,40)(39,42)(41,45)(44,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,37,39,38)(34,42,40,35)(36,45,47,46)(41,48,44,43);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,38,39,37)(34,35,40,42)(36,46,47,45)(41,43,44,48)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388667);