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 | |
4228 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)(18,24)(19,20)(25,29)(26,28)(33,34,38,43)(35,42,46,39)(36,40,41,37)(44,47,48,45);(1,2)(3,7)(4,13)(5,9)(6,11)(8,12)(10,14)(15,16)(17,18,23,24)(19,22,20,27)(21,29,31,25)(26,32,28,30)(33,35)(34,39)(36,44)(37,47)(38,46)(40,45)(41,48)(42,43);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,16)(13,15)(17,19,23,20)(18,22,24,27)(21,26,31,28)(25,30,29,32)(33,36,38,41)(34,40,43,37)(35,44,46,48)(39,45,42,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,20,23,19)(18,22,24,27)(21,28,31,26)(25,30,29,32)(33,34)(35,39)(36,40)(37,41)(38,43)(42,46)(44,45)(47,48);(1,4)(2,8)(3,10)(5,12)(6,14)(7,15)(9,13)(11,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,29)(27,32)(33,37,38,40)(34,41,43,36)(35,45,46,47)(39,44,42,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,22)(18,20)(19,24)(21,30)(23,27)(25,28)(26,29)(31,32)(33,35)(34,42)(36,44)(37,45)(38,46)(39,43)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=gh,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, 8380099);