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 | |
4356 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)(3,9,11,5)(4,8,12,14)(10,15,16,13)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34,38,40)(35,39,43,46)(36,37,44,45)(41,42,47,48);(2,7)(5,9)(8,14)(13,15)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35,38,43)(34,39,40,46)(36,41,44,47)(37,42,45,48);(4,12)(8,14)(10,16)(13,15)(18,25)(19,20)(26,32)(27,28)(34,40)(36,44)(39,46)(41,47);(1,3)(2,5)(4,10)(6,11)(7,9)(8,13)(12,16)(14,15)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,36)(34,37)(35,41)(38,44)(39,42)(40,45)(43,47)(46,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,29,24,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5)(2,3)(4,13)(6,9)(7,11)(8,10)(12,15)(14,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,37)(34,36)(35,42)(38,45)(39,41)(40,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=i,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9197723);