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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,14)(10,16)(13,15)(17,18,22,26)(19,23,20,24)(21,29,30,25)(27,32,28,31)(33,34,39,42)(35,36,37,40)(38,46,47,41)(43,48,45,44);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,48,46,44);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,32)(28,31)(29,30)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,22,19)(18,24,26,23)(21,28,30,27)(25,32,29,31)(33,37,39,35)(34,36,42,40)(38,45,47,43)(41,44,46,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)(33,38,39,47)(34,41,42,46)(35,43,37,45)(36,44,40,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,d^2=h,e^2=g,f^2=g,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, 8064152);