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 | #53406 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3984 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)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,39,43)(35,38,45,47)(36,42,46,40)(37,41,44,48);(1,2,6,7)(3,5,9,14)(4,11,13,8)(10,16,15,12)(18,19)(23,25)(24,26)(30,31)(33,35,39,45)(34,38,43,47)(36,37,46,44)(40,48,42,41);(2,7)(5,14)(8,11)(12,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,36,39,46)(34,40,43,42)(35,44,45,37)(38,41,47,48);(1,3,6,9)(2,5,7,14)(4,10,13,15)(8,12,11,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,37)(34,41)(35,36)(38,42)(39,44)(40,47)(43,48)(45,46);(1,4,6,13)(2,8,7,11)(3,10,9,15)(5,12,14,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,38)(34,35)(36,41)(37,42)(39,47)(40,44)(43,45)(46,48);(1,5)(2,9)(3,7)(4,12)(6,14)(8,15)(10,11)(13,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,36)(34,42)(35,37)(38,41)(39,46)(40,43)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8135724);