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 | |
4146 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)(3,11)(4,13)(5,15)(6,7)(8,12)(9,14)(10,16)(19,25)(23,26)(27,31)(30,32)(33,34,39,43)(35,44,41,37)(36,40,38,42)(45,46,48,47);(2,7)(5,16)(8,11)(9,13)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,35)(34,37)(36,45)(38,48)(39,41)(40,47)(42,46)(43,44);(2,7)(3,12)(9,13)(10,15)(18,22)(23,26)(24,28)(30,32)(33,36,39,38)(34,40,43,42)(35,45,41,48)(37,47,44,46);(1,3)(2,8)(4,10)(5,13)(6,12)(7,11)(9,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,37,39,44)(34,41,43,35)(36,46,38,47)(40,45,42,48);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,46)(47,48);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38,39,36)(34,42,43,40)(35,45,41,48)(37,47,44,46)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659372);