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 | #2613 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,22,24)(19,20,26,28)(21,23,32,31)(25,27,29,30)(33,34,40,43)(35,42,39,37)(36,45,38,41)(44,48,47,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,20)(21,29)(22,26)(23,30)(24,28)(25,32)(27,31)(33,35,40,39)(34,37,43,42)(36,44,38,47)(41,48,45,46);(3,10)(6,7)(11,16)(14,15)(17,20,22,28)(18,19,24,26)(21,27,32,30)(23,25,31,29)(33,36,40,38)(34,41,43,45)(35,44,39,47)(37,48,42,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,19)(21,30)(22,28)(23,29)(24,26)(25,31)(27,32)(33,37,40,42)(34,35,43,39)(36,46,38,48)(41,47,45,44);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21)(18,23)(19,25)(20,27)(22,32)(24,31)(26,29)(28,30)(33,38,40,36)(34,41,43,45)(35,44,39,47)(37,46,42,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,18)(19,20)(21,31)(22,24)(23,32)(25,30)(26,28)(27,29)(33,39,40,35)(34,42,43,37)(36,47,38,44)(41,46,45,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9074007);