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 | |
4223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,7)(4,9)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34)(35,45)(36,42)(37,38)(39,44)(40,47)(41,48)(43,46);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35)(34,40)(36,37)(38,41)(39,47)(42,43)(44,45)(46,48);(3,12)(7,8)(13,16)(14,15)(20,28)(25,30)(27,29)(31,32)(33,36,39,48)(34,41,44,42)(35,37,47,46)(38,45,43,40);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,36)(34,42)(35,46)(37,47)(38,40)(39,48)(41,44)(43,45);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,21,23,19)(18,24,22,26)(20,29,28,27)(25,31,30,32)(33,37,39,46)(34,43,44,38)(35,36,47,48)(40,42,45,41);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,22,23,18)(19,26,21,24)(20,25,28,30)(27,31,29,32)(33,38,39,43)(34,37,44,46)(35,41,47,42)(36,45,48,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653512);