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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,5)(4,8)(9,13)(12,14)(17,18)(19,26)(20,23)(21,28)(22,25)(24,30)(27,29)(31,32)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,2,7,5)(3,8,11,4)(6,9,15,13)(10,14,16,12)(18,20)(19,25)(24,28)(27,31)(34,35)(39,41)(40,42)(46,47);(1,3,7,11)(2,8,5,4)(6,10,15,16)(9,14,13,12)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4)(2,3)(5,11)(6,14)(7,8)(9,16)(10,13)(12,15)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5,7,2)(3,4,11,8)(6,13,15,9)(10,12,16,14)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32);(1,6)(2,9)(3,10)(4,12)(5,13)(7,15)(8,14)(11,16)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=gh,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8370391);