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 | #51046 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3937 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)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(1,3,6,9)(2,7,5,11)(4,10,12,13)(8,15,14,16)(17,18,23,27)(19,28,30,24)(20,22,26,32)(21,25,29,31)(34,40)(35,44)(37,47)(42,45);(2,5)(3,9)(4,12)(15,16)(17,19,23,30)(18,24,27,28)(20,21,26,29)(22,31,32,25)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,13)(11,16)(17,20,23,26)(18,22,27,32)(19,29,30,21)(24,25,28,31)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,12)(14,15)(17,21,23,29)(18,25,27,31)(19,20,30,26)(22,24,32,28)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,5,6,2)(3,11,9,7)(4,14,12,8)(10,16,13,15)(17,22)(18,26)(19,25)(20,27)(21,24)(23,32)(28,29)(30,31)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=ghi,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8924723);