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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3918 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)(3,5)(4,12)(6,7)(8,11)(9,13)(10,15)(14,16)(19,25)(22,26)(27,31)(30,32)(33,34,38,39)(35,42,40,36)(37,41,45,46)(43,48,47,44);(2,7)(4,13)(5,8)(10,16)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(34,39)(35,40)(37,45)(44,48);(1,3,6,11)(2,5,7,8)(4,10,13,16)(9,14,12,15)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,35,38,40)(34,36,39,42)(37,43,45,47)(41,44,46,48);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,46)(45,48);(1,4,6,13)(2,9,7,12)(3,10,11,16)(5,14,8,15)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5,6,8)(2,3,7,11)(4,14,13,15)(9,10,12,16)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,46)(45,48)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,e^2=gi,f^2=gh,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, 8664284);