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 | |
4062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,13,9)(4,10,15,5)(11,12,14,16)(17,18)(19,28)(20,31)(21,22)(23,24)(25,29)(26,30)(27,32)(34,39)(37,40)(41,46)(45,47);(2,7)(5,10)(8,9)(12,16)(18,24)(19,29)(21,27)(26,31)(36,44)(41,46)(43,48)(45,47);(1,3,6,13)(2,8,7,9)(4,14,15,11)(5,12,10,16)(17,19)(18,25)(20,27)(21,30)(22,31)(23,29)(24,28)(26,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3)(2,9)(4,11)(5,12)(6,13)(7,8)(10,16)(14,15)(17,20,23,30)(18,26,24,31)(19,27,29,21)(22,25,32,28)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,4)(2,10)(3,11)(5,7)(6,15)(8,12)(9,16)(13,14)(17,21,23,27)(18,22,24,32)(19,30,29,20)(25,31,28,26)(33,36,38,44)(34,41,39,46)(35,43,42,48)(37,45,40,47);(1,5,6,10)(2,4,7,15)(3,12,13,16)(8,14,9,11)(17,22)(18,27)(19,31)(20,25)(21,24)(23,32)(26,29)(28,30)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)Polycyclic presentation:
2^9|a^2=gi,c^2=gi,d^2=h,e^2=h,f^2=gh,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, 8679826);