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 | #46052 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(19,26)(22,24)(27,32)(29,31)(33,34)(35,37)(36,44)(38,42)(39,41)(40,46)(43,47)(45,48);(2,7)(4,13)(5,8)(10,16)(17,18,23,20)(19,24,26,22)(21,28,30,25)(27,29,32,31)(33,34,38,42)(35,37,41,39)(36,40,46,44)(43,45,48,47);(1,3,6,11)(2,5,7,8)(4,10,13,16)(9,14,12,15)(17,19,23,26)(18,24,20,22)(21,27,30,32)(25,31,28,29)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,47,44,45);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,14)(13,16)(17,20,23,18)(19,22,26,24)(21,28,30,25)(27,29,32,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,4,6,13)(2,9,7,12)(3,10,11,16)(5,14,8,15)(17,21,23,30)(18,25,20,28)(19,27,26,32)(22,29,24,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,46)(45,48);(1,5,6,8)(2,3,7,11)(4,14,13,15)(9,10,12,16)(17,22)(18,19)(20,26)(21,29)(23,24)(25,27)(28,32)(30,31)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,48,44,43)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=hi,e^2=gh,f^2=g,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, 8662609);