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 | |
4078 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(18,22)(19,25)(20,28)(30,32)(33,34)(35,36)(37,39)(38,45)(40,42)(41,46)(43,48)(44,47);(2,7)(3,10)(8,14)(11,16)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35,39,40)(34,36,37,42)(38,43,46,47)(41,44,45,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,36,39,42)(34,40,37,35)(38,44,46,48)(41,47,45,43);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36)(34,35)(37,40)(38,44)(39,42)(41,43)(45,47)(46,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,37,39,34)(35,42,40,36)(38,45,46,41)(43,48,47,44);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,38,39,46)(34,41,37,45)(35,43,40,47)(36,44,42,48)Polycyclic presentation:
2^9|b^2=gi,c^2=gi,e^2=gh,f^2=h,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, 8066199);