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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,11)(4,13)(6,12)(8,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,2,7,10)(3,12,11,6)(4,8,13,16)(5,9,15,14)(20,28)(23,30)(26,32)(29,31)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(1,3,7,11)(2,6,10,12)(4,5,13,15)(8,14,16,9)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,4)(2,8)(3,5)(6,14)(7,13)(9,12)(10,16)(11,15)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,5)(2,9)(3,13)(4,11)(6,8)(7,15)(10,14)(12,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,6)(2,3)(4,14)(5,8)(7,12)(9,13)(10,11)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833736);