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 | #40963 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(33,34,39,40)(35,37,42,38)(36,44,47,41)(43,46,48,45);(1,2,6,7)(3,8,13,16)(4,9,14,12)(5,10,15,11)(18,19)(23,25)(24,26)(30,31)(34,40)(35,42)(41,44)(43,48);(2,7)(4,14)(8,16)(10,11)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35)(34,37)(36,43)(38,40)(39,42)(41,45)(44,46)(47,48);(1,3)(2,8)(4,11)(5,9)(6,13)(7,16)(10,14)(12,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,4)(2,9)(3,11)(5,8)(6,14)(7,12)(10,13)(15,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37,39,38)(34,42,40,35)(36,45,47,46)(41,48,44,43);(1,5)(2,10)(3,12)(4,8)(6,15)(7,11)(9,13)(14,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38,39,37)(34,35,40,42)(36,46,47,45)(41,43,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8424784);