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 | #43373 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,24)(19,27,29,23)(20,25,30,31)(21,32,28,26)(33,34)(35,39)(36,40)(37,43)(38,41)(42,47)(44,46)(45,48);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,15)(14,16)(17,19,22,29)(18,23,24,27)(20,28,30,21)(25,32,31,26)(33,35,38,43)(34,39,41,37)(36,42,46,48)(40,47,44,45);(2,6)(3,10)(4,13)(15,16)(18,24)(20,30)(21,28)(23,27)(33,34,38,41)(35,37,43,39)(36,44,46,40)(42,47,48,45);(1,3)(2,7)(4,14)(5,10)(6,9)(8,16)(11,13)(12,15)(17,18)(19,27)(20,31)(21,26)(22,24)(23,29)(25,30)(28,32)(33,34)(35,37)(36,44)(38,41)(39,43)(40,46)(42,47)(45,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,15)(9,16)(10,14)(17,20)(18,25)(19,28)(21,29)(22,30)(23,32)(24,31)(26,27)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,21,22,28)(18,26,24,32)(19,20,29,30)(23,25,27,31)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965478);