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 | #2539 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(10,16)(13,14)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34)(35,40)(36,46)(37,41)(38,47)(39,42)(43,44)(45,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,35,39,45)(34,40,42,48)(36,37,47,43)(38,44,46,41);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,36)(34,38)(35,43)(37,45)(39,47)(40,41)(42,46)(44,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,37,39,43)(34,41,42,44)(35,36,45,47)(38,40,46,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,36)(34,38)(35,37)(39,47)(40,44)(41,48)(42,46)(43,45);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,38)(34,36)(35,44)(37,40)(39,46)(41,45)(42,47)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8042850);