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 | |
3880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,10,6,8)(5,9,13,12)(11,16,14,15)(19,25)(23,29)(26,31)(30,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,45)(43,44)(47,48);(1,3,7,6)(2,8,4,10)(5,11,13,14)(9,15,12,16)(18,22)(23,29)(24,27)(30,32)(34,38)(39,45)(40,43)(46,47);(2,4)(8,10)(9,12)(15,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(33,35,37,42)(34,39,38,45)(36,41,44,48)(40,46,43,47);(1,4,7,2)(3,10,6,8)(5,12,13,9)(11,16,14,15)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(33,34)(35,39)(36,43)(37,38)(40,44)(41,47)(42,45)(46,48);(1,6,7,3)(2,10,4,8)(5,11,13,14)(9,15,12,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,36,37,44)(34,40,38,43)(35,41,42,48)(39,46,45,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=h,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^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817883);