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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4109 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)(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(34,40)(36,44)(37,45)(41,48);(1,2)(3,7)(4,13)(5,9)(6,11)(8,12)(10,14)(15,16)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(33,34)(35,41)(36,42)(37,46)(38,44)(39,40)(43,45)(47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,16)(13,15)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,35,39,47)(34,41,40,48)(36,45,44,37)(38,46,42,43);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,36)(34,42)(35,45)(37,47)(38,40)(39,44)(41,43)(46,48);(1,4)(2,8)(3,10)(5,12)(6,14)(7,15)(9,13)(11,16)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29)(33,37,39,45)(34,43,40,46)(35,36,47,44)(38,48,42,41);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(33,38,39,42)(34,44,40,36)(35,46,47,43)(37,48,45,41)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8380129);