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 | #2847 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4048 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)(7,14)(10,16)(13,15)(17,18,22,25)(19,26,23,20)(21,29,30,24)(27,28,31,32)(33,34)(35,41)(36,45)(37,40)(38,47)(39,42)(43,44)(46,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,35,39,37)(34,40,42,41)(36,43,47,46)(38,44,45,48);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,35,39,37)(34,41,42,40)(36,46,47,43)(38,44,45,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,36)(34,38)(35,43)(37,46)(39,47)(40,44)(41,48)(42,45);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,37,39,35)(34,40,42,41)(36,43,47,46)(38,48,45,44);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38)(34,36)(35,44)(37,48)(39,45)(40,43)(41,46)(42,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064349);