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 | #57585 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,12,6)(4,10,13,16)(5,11,15,14)(17,18)(19,27)(20,23)(21,25)(22,30)(24,28)(26,31)(29,32)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(1,3,7,12)(2,8,9,6)(4,5,13,15)(10,11,16,14)(17,19,23,28)(18,24,20,27)(21,31,32,22)(25,30,29,26)(36,44)(40,47)(42,48)(45,46);(2,9)(3,12)(4,13)(11,14)(18,20)(24,27)(25,29)(26,30)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,4)(2,10)(3,5)(6,14)(7,13)(8,11)(9,16)(12,15)(17,20,23,18)(19,27,28,24)(21,29,32,25)(22,30,31,26)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,5)(2,11)(3,4)(6,16)(7,15)(8,10)(9,14)(12,13)(17,21,23,32)(18,25,20,29)(19,22,28,31)(24,26,27,30)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6,7,8)(2,12,9,3)(4,14,13,11)(5,10,15,16)(17,22)(18,26)(19,21)(20,30)(23,31)(24,25)(27,29)(28,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,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, 9619210);