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 | |
4212 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)(17,18,23,25)(19,24,26,22)(20,29,31,21)(27,28,30,32)(33,34,39,37)(35,36,41,40)(38,42,47,45)(43,44,46,48);(3,9)(7,13)(10,15)(14,16)(17,19)(18,24)(20,30)(21,28)(22,25)(23,26)(27,31)(29,32)(33,35,39,41)(34,40,37,36)(38,43,47,46)(42,48,45,44);(2,6)(7,13)(8,11)(14,16)(18,25)(19,26)(20,31)(28,32)(33,36,39,40)(34,41,37,35)(38,44,47,48)(42,46,45,43);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,21)(19,27)(22,28)(23,31)(24,32)(25,29)(26,30)(33,37,39,34)(35,40,41,36)(38,45,47,42)(43,48,46,44);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21)(18,20)(19,28)(22,27)(23,29)(24,30)(25,31)(26,32)(33,35)(34,36)(37,40)(38,46)(39,41)(42,48)(43,47)(44,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22)(18,26)(19,25)(20,28)(21,30)(23,24)(27,29)(31,32)(33,38,39,47)(34,42,37,45)(35,43,41,46)(36,44,40,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9843108);