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 | |
4159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,14)(10,15)(11,16)(17,18,22,25)(19,26,23,20)(21,29,30,24)(27,28,31,32)(33,34,38,42)(35,41,37,39)(36,40,46,44)(43,48,45,47);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(2,7)(3,9)(8,14)(10,15)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,3,6,9)(2,4,7,12)(5,10,13,15)(8,11,14,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,48)(45,47);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21,22,30)(18,24,25,29)(19,27,23,31)(20,28,26,32)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^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,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066612);