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 | #51932 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3899 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)(7,14)(10,12)(15,16)(17,18)(19,21)(20,22)(23,25)(24,28)(26,30)(27,31)(29,32)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19,23,28)(18,24,25,21)(20,26,29,31)(22,27,32,30)(35,41)(38,45)(42,47)(46,48);(2,6)(3,9)(8,11)(10,12)(20,29)(22,32)(26,31)(27,30)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,20)(18,22)(19,26)(21,30)(23,29)(24,27)(25,32)(28,31);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21,23,24)(18,19,25,28)(20,27,29,30)(22,31,32,26)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,22)(18,20)(19,27)(21,31)(23,32)(24,26)(25,29)(28,30)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837562);