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 | #56769 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3950 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)(5,15)(6,11)(7,9)(8,13)(10,12)(14,16)(17,18,23,21)(19,27,30,24)(20,25,29,31)(22,26,28,32)(33,34)(35,42)(36,37)(38,40)(39,43)(41,48)(44,45)(46,47);(1,3,7,13)(2,8,9,4)(5,6,12,16)(10,11,15,14)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,26)(31,32)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(2,9)(4,8)(10,15)(11,14)(17,20,23,29)(18,25,21,31)(19,28,30,22)(24,26,27,32)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,4,7,8)(2,3,9,13)(5,14,12,11)(6,15,16,10)(17,20)(18,25)(19,28)(21,31)(22,30)(23,29)(24,26)(27,32)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,5)(2,10)(3,6)(4,14)(7,12)(8,11)(9,15)(13,16)(17,21,23,18)(19,27,30,24)(20,31,29,25)(22,26,28,32)(33,35,39,38)(34,42,43,40)(36,44,41,46)(37,45,48,47);(1,6)(2,11)(3,12)(4,10)(5,13)(7,16)(8,15)(9,14)(17,22,23,28)(18,26,21,32)(19,29,30,20)(24,25,27,31)(33,38,39,35)(34,40,43,42)(36,46,41,44)(37,47,48,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 8646639);