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 | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3963 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)(3,4)(5,13)(6,8)(7,11)(9,15)(10,16)(12,14)(18,23)(19,26)(21,29)(28,31)(33,34,38,40)(35,36,42,44)(37,39,47,46)(41,43,45,48);(1,3,6,11)(2,7,8,4)(5,10,15,14)(9,16,13,12)(17,18,22,23)(19,20,26,24)(21,25,29,30)(27,28,32,31)(33,35)(34,36)(37,45)(38,42)(39,48)(40,44)(41,47)(43,46);(2,8)(4,7)(9,13)(12,16)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,36,38,44)(34,35,40,42)(37,43,47,48)(39,41,46,45);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,14)(15,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,46)(47,48);(1,3)(2,4)(5,14)(6,11)(7,8)(9,16)(10,15)(12,13)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,31)(30,32)(33,37)(34,39)(35,41)(36,43)(38,47)(40,46)(42,45)(44,48);(1,5)(2,9)(3,10)(4,12)(6,15)(7,16)(8,13)(11,14)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)(33,34)(35,36)(37,46)(38,40)(39,47)(41,48)(42,44)(43,45)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9627595);