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 | |
3982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,14)(10,16)(13,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,31)(29,32)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,2)(3,8)(4,7)(5,9)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,25)(26,27)(31,32)(34,38)(39,41)(40,45)(46,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,22,29)(18,24,26,32)(19,28,30,21)(23,25,27,31)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4)(2,7)(3,10)(5,12)(6,13)(8,15)(9,14)(11,16)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,21,22,28)(18,25,26,31)(19,29,30,20)(23,24,27,32);(1,2)(3,6)(4,7)(5,9)(8,11)(10,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,28)(27,30)(29,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8005019);