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 | |
3743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,11,5,9)(4,10,14,6)(12,16,15,13)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,37,40)(35,38,42,46)(36,43,44,39)(41,47,48,45);(2,8)(4,14)(9,11)(12,15)(18,19)(20,27)(24,26)(28,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(1,3,7,5)(2,9,8,11)(4,15,14,12)(6,13,10,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,7,14)(2,10,8,6)(3,12,5,15)(9,13,11,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5,7,3)(2,11,8,9)(4,15,14,12)(6,13,10,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47);(1,6,7,10)(2,4,8,14)(3,13,5,16)(9,15,11,12)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,d^2=gi,e^2=h,f^2=gi,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, 8657988);