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 | |
3741 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)(3,10)(4,6)(5,9)(7,8)(11,12)(13,16)(14,15)(19,27)(20,29)(23,30)(24,32)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(2,8)(4,13)(9,10)(11,14)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,28,31)(34,39)(35,41)(40,46)(42,47);(1,3,7,5)(2,9,8,10)(4,14,13,11)(6,12,16,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,32)(26,30)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,4,7,13)(2,6,8,16)(3,11,5,14)(9,15,10,12)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,5,7,3)(2,10,8,9)(4,14,13,11)(6,12,16,15)(17,20,22,29)(18,24,26,32)(19,28,27,21)(23,31,30,25);(1,6)(2,4)(3,12)(5,15)(7,16)(8,13)(9,14)(10,11)(17,21,22,28)(18,25,26,31)(19,29,27,20)(23,32,30,24)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gi,d^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653198);