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 | #58071 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,6,11,8)(4,12,13,9)(10,15,16,14)(17,18)(19,20)(21,23)(22,29)(24,26)(25,30)(27,32)(28,31)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(1,3,7,11)(2,8,5,6)(4,10,13,16)(9,15,12,14)(17,19,23,24)(18,20,21,26)(22,27,30,31)(25,28,29,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(4,13)(9,12)(10,16)(14,15)(17,20,23,26)(18,24,21,19)(22,28,30,32)(25,31,29,27)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,4)(2,9)(3,10)(5,12)(6,14)(7,13)(8,15)(11,16)(17,20)(18,19)(21,24)(22,28)(23,26)(25,27)(29,31)(30,32)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(1,5,7,2)(3,6,11,8)(4,9,13,12)(10,15,16,14)(17,21,23,18)(19,26,24,20)(22,29,30,25)(27,32,31,28)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,6)(2,3)(4,14)(5,11)(7,8)(9,10)(12,16)(13,15)(17,22,23,30)(18,25,21,29)(19,27,24,31)(20,28,26,32)(33,37,38,46)(34,40,41,45)(35,43,39,47)(36,44,42,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=i,e^2=gh,f^2=h,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, 8066196);