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 | |
4356 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,23)(19,20,24,27)(21,25,30,31)(26,28,32,29)(33,34)(35,40)(36,45)(37,42)(38,46)(39,41)(43,48)(44,47);(3,9)(7,13)(10,15)(14,16)(18,23)(20,27)(25,31)(28,29)(33,35)(34,40)(36,37)(38,44)(39,43)(41,48)(42,45)(46,47);(2,6)(7,13)(8,11)(14,16)(34,41)(35,43)(36,46)(42,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,24)(18,20,23,27)(21,26,30,32)(25,28,31,29)(33,36)(34,38)(35,37)(39,46)(40,44)(41,45)(42,48)(43,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20)(18,24)(19,23)(21,29)(22,27)(25,26)(28,30)(31,32)(33,37)(34,42)(35,36)(38,48)(39,47)(40,45)(41,44)(43,46);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,38)(34,36)(35,44)(37,40)(39,45)(41,46)(42,43)(47,48)Polycyclic presentation:
2^9|a^2=g,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817016);