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 | #2813 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3826 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(18,22)(20,27)(23,28)(26,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,7)(3,8)(9,16)(10,13)(19,25)(23,28)(26,31)(30,32)(34,35)(39,41)(40,42)(46,47);(2,7)(3,8)(9,16)(10,13)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gi,e^2=gh,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, 8658470);