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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(19,26)(22,24)(27,32)(29,31)(33,34,39,44)(35,45,47,40)(36,43,38,41)(37,42,46,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,18,23,20)(19,24,26,22)(21,28,30,25)(27,29,32,31)(33,34)(35,45)(36,41)(37,48)(38,43)(39,44)(40,47)(42,46);(2,6)(7,10)(8,12)(14,16)(17,19,23,26)(18,24,20,22)(21,27,30,32)(25,31,28,29)(33,35,39,47)(34,40,44,45)(36,46,38,37)(41,48,43,42);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,22,26,24)(21,28,30,25)(27,29,32,31)(33,36,39,38)(34,41,44,43)(35,46,47,37)(40,48,45,42);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37)(34,42)(35,36)(38,47)(39,46)(40,41)(43,45)(44,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,19)(20,26)(21,29)(23,24)(25,27)(28,32)(30,31)(33,38,39,36)(34,43,44,41)(35,37,47,46)(40,42,45,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659414);