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 | #50933 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3833 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,14)(15,16)(18,22)(23,28)(24,29)(30,32)(33,34,39,37)(35,42,41,38)(36,44,46,40)(43,45,47,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(20,27)(24,29)(25,31)(30,32)(34,37)(35,41)(40,44)(43,47);(1,2,5,8)(3,6,11,9)(4,7,13,12)(10,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35)(34,38)(36,43)(37,42)(39,41)(40,45)(44,48)(46,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)(33,37,39,34)(35,42,41,38)(36,44,46,40)(43,48,47,45);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,38,39,42)(34,41,37,35)(36,45,46,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662378);