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 | |
4278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,10,7)(4,8,12,13)(11,15,16,14)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,34)(35,42)(36,44)(37,40)(38,39)(41,46)(43,47)(45,48);(2,5)(7,9)(8,13)(14,15)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,48)(45,46);(3,10)(4,12)(7,9)(8,13)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,34,39,38)(35,40,37,42)(36,44,46,41)(43,48,45,47);(1,3,6,10)(2,7,5,9)(4,11,12,16)(8,14,13,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,36,39,46)(34,41,38,44)(35,43,37,45)(40,47,42,48);(1,4,6,12)(2,8,5,13)(3,11,10,16)(7,14,9,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,37,39,35)(34,40,38,42)(36,45,46,43)(41,47,44,48);(1,5,6,2)(3,9,10,7)(4,13,12,8)(11,15,16,14)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,38,39,34)(35,40,37,42)(36,44,46,41)(43,47,45,48)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669246);