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 | #45720 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,9,5,7)(4,8,12,15)(11,16,13,14)(17,18)(19,21)(20,28)(22,24)(23,26)(25,31)(27,32)(29,30)(36,44)(40,45)(42,48)(46,47);(1,3,6,5)(2,7,10,9)(4,11,12,13)(8,14,15,16)(17,19,22,26)(18,23,24,21)(20,29,31,27)(25,30,28,32)(34,38)(39,41)(40,45)(46,47);(4,12)(8,15)(11,13)(14,16)(18,24)(19,26)(25,28)(27,29)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,20,22,31)(18,25,24,28)(19,27,26,29)(21,30,23,32);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,32)(28,31)(29,30)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5,6,3)(2,9,10,7)(4,13,12,11)(8,16,15,14)(17,21,22,23)(18,26,24,19)(20,30,31,32)(25,29,28,27)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=hi,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002358);