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 | |
4220 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)(3,10)(4,8)(5,7)(6,13)(9,14)(11,15)(12,16)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(3,4)(8,10)(11,12)(15,16)(18,22)(23,25)(24,28)(30,31)(34,39)(35,42)(41,44)(43,48);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4,7,3)(2,8,5,10)(6,12,14,11)(9,15,13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,7,2)(3,8,4,10)(6,13,14,9)(11,15,12,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026831);