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 | #39776 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,11)(8,13)(9,15)(14,16)(17,18)(19,24)(20,27)(21,29)(22,23)(25,30)(26,28)(31,32)(33,34,39,42)(35,41,36,40)(37,38,47,48)(43,46,45,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,18)(19,20)(21,25)(22,23)(24,27)(26,28)(29,30)(31,32)(33,35,39,36)(34,40,42,41)(37,43,47,45)(38,44,48,46);(2,6)(7,12)(8,13)(14,16)(18,23)(20,24)(25,29)(28,31)(33,35,39,36)(34,41,42,40)(37,45,47,43)(38,44,48,46);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,29)(30,32)(33,36,39,35)(34,41,42,40)(37,45,47,43)(38,46,48,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,20,22,24)(18,19,23,27)(21,28,30,31)(25,26,29,32)(33,37)(34,38)(35,43)(36,45)(39,47)(40,44)(41,46)(42,48);(1,4)(2,8)(3,9)(5,11)(6,13)(7,14)(10,15)(12,16)(17,21,22,30)(18,25,23,29)(19,26,27,32)(20,28,24,31)(33,38)(34,37)(35,44)(36,46)(39,48)(40,43)(41,45)(42,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8025862);