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 | #40806 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4155 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)(8,13)(10,16)(14,15)(17,18,22,25)(19,26,23,20)(21,29,30,24)(27,28,31,32)(33,34,39,42)(35,41,38,40)(36,37,46,48)(43,47,45,44);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,35,39,38)(34,40,42,41)(36,43,46,45)(37,44,48,47);(2,6)(7,9)(8,13)(14,15)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,36)(34,37)(35,43)(38,45)(39,46)(40,44)(41,47)(42,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,37,39,48)(34,36,42,46)(35,44,38,47)(40,43,41,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,36)(34,37)(35,43)(38,45)(39,46)(40,44)(41,47)(42,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,30)(18,24,25,29)(19,27,23,31)(20,28,26,32)(33,38,39,35)(34,41,42,40)(36,45,46,43)(37,47,48,44)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=gh,f^2=g,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, 8062236);