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 |
| prime radical | #2671 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3959 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)(3,8)(4,13)(5,10)(6,7)(9,14)(11,12)(15,16)(17,18,23,28)(19,29,31,24)(20,22,27,25)(21,32,30,26)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(1,2)(3,11)(4,9)(5,10)(6,7)(8,12)(13,14)(15,16)(17,19)(18,24)(20,21)(22,26)(23,31)(25,32)(27,30)(28,29)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(2,7)(3,12)(9,13)(10,16)(17,20,23,27)(18,25,28,22)(19,21,31,30)(24,32,29,26)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,3)(2,8)(4,10)(5,13)(6,12)(7,11)(9,15)(14,16)(17,20)(18,25)(19,30)(21,31)(22,28)(23,27)(24,26)(29,32)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,15)(12,16)(17,21,23,30)(18,26,28,32)(19,20,31,27)(22,29,25,24)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,5,6,15)(2,10,7,16)(3,9,12,13)(4,8,14,11)(17,22,23,25)(18,27,28,20)(19,26,31,32)(21,24,30,29)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627594);