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 | #55238 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3842 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,23)(19,25,27,20)(21,29,31,24)(26,28,30,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(2,6)(7,9)(8,13)(14,15)(17,19,22,27)(18,20,23,25)(21,30,31,26)(24,32,29,28)(35,43)(37,47)(39,48)(41,44);(4,12)(8,13)(10,16)(14,15)(18,23)(20,25)(24,29)(28,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,31)(30,32);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,30)(31,32)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,21,22,31)(18,24,23,29)(19,26,27,30)(20,28,25,32)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9148146);