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 | |
4938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,33)(28,32)(29,34)(30,31)(35,40)(36,37)(38,39);(1,3)(2,5)(4,8)(6,7)(11,17)(15,21)(18,23)(22,24)(25,27)(26,29)(28,36)(30,34)(31,33)(32,40)(35,38)(37,39);(10,14)(15,21)(16,19)(22,24)(26,31)(27,34)(32,39)(35,36);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,2)(3,5)(4,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,39)(38,40)Polycyclic presentation:
2^9|e^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219217, #72219279, #72219283 |
| maximal quotient of | #72219214 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9621752);