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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,13,15)(4,5,10,9)(11,16,14,12)(17,18)(19,26)(20,23)(21,28)(22,25)(24,30)(27,29)(31,32)(33,34,39,42)(35,36,38,41)(37,40,46,48)(43,44,45,47);(2,7)(4,10)(8,15)(11,14)(18,20)(19,25)(24,28)(27,31)(33,35,39,38)(34,36,42,41)(37,45,46,43)(40,47,48,44);(2,7)(5,9)(8,15)(12,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,36,39,41)(34,35,42,38)(37,44,46,47)(40,43,48,45);(1,3)(2,8)(4,14)(5,12)(6,13)(7,15)(9,16)(10,11)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29)(33,37,39,46)(34,40,42,48)(35,43,38,45)(36,44,41,47);(1,4,6,10)(2,9,7,5)(3,11,13,14)(8,12,15,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,5,6,9)(2,10,7,4)(3,12,13,16)(8,11,15,14)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,38,39,35)(34,41,42,36)(37,45,46,43)(40,47,48,44)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324331);