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 | #2416 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3908 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)(8,13)(10,16)(14,15)(17,18)(19,26)(20,25)(21,30)(22,23)(24,29)(27,28)(31,32)(33,34,39,41)(35,40,36,44)(37,38,47,48)(42,46,45,43);(1,2)(3,7)(4,13)(5,9)(6,11)(8,12)(10,14)(15,16)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,31)(30,32)(33,35,39,36)(34,40,41,44)(37,45,47,42)(38,43,48,46);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,16)(13,15)(17,20,23,30)(18,25,22,21)(19,27,29,32)(24,31,26,28)(33,36,39,35)(34,40,41,44)(37,45,47,42)(38,46,48,43);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,21)(18,20)(19,28)(22,30)(23,25)(24,27)(26,32)(29,31)(33,37)(34,38)(35,42)(36,45)(39,47)(40,46)(41,48)(43,44);(1,4)(2,8)(3,10)(5,12)(6,14)(7,15)(9,13)(11,16)(17,22,23,18)(19,24,29,26)(20,21,30,25)(27,31,32,28)(33,38)(34,37)(35,43)(36,46)(39,48)(40,45)(41,47)(42,44);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,19)(18,26)(20,27)(21,31)(22,24)(23,29)(25,28)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8378855);