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 | |
4413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(25,26,30,33)(27,34,29,31)(28,36,38,32)(35,39,37,40);(3,8)(6,7)(9,10,14,15)(11,12,16,20)(13,17,21,22)(18,19,23,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(10,15)(11,16)(13,21)(19,24)(25,27,30,29)(26,31,33,34)(28,37,38,35)(32,40,36,39);(3,8)(6,7)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,23)(22,24)(25,28,30,38)(26,32,33,36)(27,35,29,37)(31,39,34,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29,30,27)(26,31,33,34)(28,35,38,37)(32,40,36,39);(1,4,5,2)(3,6,8,7)(9,13)(10,17)(11,18)(12,19)(14,21)(15,22)(16,23)(20,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219279, #72219281, #72219283 |
| maximal quotient of | #72219272 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8816967);