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 | |
4123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,5,9)(6,12,14,10)(11,16,13,15)(33,34,39,43)(35,44,46,40)(36,38,47,41)(37,42,48,45);(1,3,7,5)(2,8,4,9)(6,11,14,13)(10,15,12,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,35,39,46)(34,40,43,44)(36,42,47,45)(37,41,48,38);(2,4)(3,5)(6,14)(15,16)(18,22)(19,26)(20,29)(31,32)(33,36,39,47)(34,41,43,38)(35,42,46,45)(37,40,48,44);(1,4,7,2)(3,8,5,9)(6,12,14,10)(11,15,13,16)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,36)(34,41)(35,45)(37,40)(38,43)(39,47)(42,46)(44,48);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,37)(34,42)(35,41)(36,44)(38,46)(39,48)(40,47)(43,45);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38,39,41)(34,36,43,47)(35,37,46,48)(40,42,44,45)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,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, 8966464);