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 |
| prime radical | #48826 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,18)(14,16)(15,17)(19,20)(21,23)(22,24);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,16)(13,21)(14,17)(18,20)(19,22)(23,24)(25,26)(27,32)(28,29)(30,31)(33,38)(34,40)(35,37)(36,39);(1,2,5,4)(3,6,8,7)(10,16)(11,17)(13,22)(20,23)(26,31)(29,38)(32,39)(35,40);(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,22)(23,24)(25,27)(26,32)(28,37)(29,35)(30,36)(31,39)(33,34)(38,40);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,16)(13,20)(14,15)(18,21)(19,24)(22,23)(25,28,30,33)(26,29,31,38)(27,34,36,37)(32,40,39,35);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,18,16,24)(11,19,17,21)(12,20,15,23)(25,29)(26,33)(27,35)(28,31)(30,38)(32,34)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7733904);