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 | |
4249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,9,4,13)(5,14,15,10)(6,11,16,12)(20,28)(24,29)(26,32)(30,31)(34,39)(35,41)(42,45)(43,48);(2,8)(5,15)(9,13)(11,12)(17,18)(19,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32)(33,34,38,39)(35,40,41,36)(37,42,46,45)(43,47,48,44);(1,3,7,4)(2,9,8,13)(5,12,15,11)(6,10,16,14)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,4,7,3)(2,9,8,13)(5,11,15,12)(6,10,16,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,44)(47,48);(1,5)(2,10)(3,12)(4,11)(6,13)(7,15)(8,14)(9,16)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,6,7,16)(2,11,8,12)(3,10,4,14)(5,13,15,9)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,37,38,46)(34,42,39,45)(35,43,41,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8380122);