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 | |
4127 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)(3,7)(4,5)(6,9)(8,12)(10,11)(13,16)(14,15)(17,18,22,25)(19,26,23,20)(21,24,31,32)(27,30,29,28)(33,34,38,41)(35,42,37,39)(36,44,46,40)(43,47,45,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,9)(13,14)(15,16)(17,19)(18,20)(21,29)(22,23)(24,30)(25,26)(27,31)(28,32)(35,37)(39,42)(43,45)(47,48);(1,3,6,12)(2,7,9,8)(4,10,14,13)(5,11,15,16)(17,20,22,26)(18,23,25,19)(21,28,31,30)(24,29,32,27)(33,35,38,37)(34,39,41,42)(36,45,46,43)(40,48,44,47);(1,3)(2,8)(4,13)(5,11)(6,12)(7,9)(10,14)(15,16)(17,20)(18,19)(21,30)(22,26)(23,25)(24,29)(27,32)(28,31)(33,36,38,46)(34,40,41,44)(35,43,37,45)(39,47,42,48);(1,4,6,14)(2,5,9,15)(3,10,12,13)(7,11,8,16)(17,21)(18,24)(19,27)(20,28)(22,31)(23,29)(25,32)(26,30)(33,37,38,35)(34,39,41,42)(36,43,46,45)(40,48,44,47);(1,5)(2,4)(3,11)(6,15)(7,10)(8,13)(9,14)(12,16)(17,19)(18,20)(21,29)(22,23)(24,30)(25,26)(27,31)(28,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113968);