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 | |
4133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,12)(5,10)(6,7)(9,13)(11,15)(14,16)(17,18,22,24)(19,23,27,32)(20,28,30,25)(21,31,29,26)(35,41)(36,44)(39,42)(40,45);(2,7)(3,10)(4,13)(15,16)(17,19,22,27)(18,23,24,32)(20,29,30,21)(25,31,28,26)(33,34)(35,42)(36,45)(37,38)(39,41)(40,44)(43,47)(46,48);(3,10)(5,8)(11,14)(15,16)(18,24)(19,27)(20,30)(26,31)(34,38)(35,41)(40,45)(43,46);(1,3,6,10)(2,8,7,5)(4,14,13,11)(9,15,12,16)(17,20,22,30)(18,25,24,28)(19,21,27,29)(23,26,32,31)(33,35)(34,39)(36,46)(37,41)(38,42)(40,48)(43,44)(45,47);(1,4)(2,9)(3,11)(5,15)(6,13)(7,12)(8,16)(10,14)(17,21)(18,26)(19,20)(22,29)(23,25)(24,31)(27,30)(28,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,47)(41,46)(42,48);(1,5,6,8)(2,10,7,3)(4,15,13,16)(9,14,12,11)(17,18)(19,23)(20,25)(21,26)(22,24)(27,32)(28,30)(29,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840363);