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 | |
4167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,10,8)(4,12,13,9)(11,15,16,14)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34,39,41)(35,40,44,48)(36,38,43,46)(37,45,47,42);(2,7)(3,10)(4,13)(14,15)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,35,39,44)(34,40,41,48)(36,42,43,45)(37,46,47,38);(2,7)(5,8)(9,12)(14,15)(21,29)(24,31)(26,32)(28,30)(34,41)(35,44)(38,46)(42,45);(1,3,6,10)(2,8,7,5)(4,11,13,16)(9,15,12,14)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,36,39,43)(34,38,41,46)(35,45,44,42)(37,40,47,48);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,37)(34,42)(35,38)(36,40)(39,47)(41,45)(43,48)(44,46);(1,5,6,8)(2,10,7,3)(4,14,13,15)(9,16,12,11)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,38)(34,43)(35,37)(36,41)(39,46)(40,45)(42,48)(44,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109396);