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 | #40943 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,12,5)(4,8,14,16)(10,15,11,13)(18,20)(21,29)(23,25)(26,30)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(1,3,6,12)(2,7,9,5)(4,11,14,10)(8,13,16,15)(18,20)(23,25)(24,28)(31,32)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,46)(47,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,18,22,20)(19,23,27,25)(21,24,29,28)(26,31,30,32)(33,34,38,41)(35,36,39,42)(37,40,46,48)(43,44,45,47);(1,4)(2,8)(3,11)(5,15)(6,14)(7,13)(9,16)(10,12)(17,19,22,27)(18,23,20,25)(21,30,29,26)(24,32,28,31)(33,35)(34,36)(37,45)(38,39)(40,47)(41,42)(43,46)(44,48);(1,3)(2,5)(4,10)(6,12)(7,9)(8,13)(11,14)(15,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,32,30,31)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,5)(2,3)(4,13)(6,7)(8,10)(9,12)(11,16)(14,15)(17,21,22,29)(18,24,20,28)(19,26,27,30)(23,31,25,32)(33,37,38,46)(34,40,41,48)(35,43,39,45)(36,44,42,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10330066);