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 | #38025 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(13,21)(16,23)(19,22)(20,24)(25,26,30,34)(27,33,29,31)(28,32,38,36)(35,40,37,39);(1,3,5,8)(2,6,4,7)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,38)(37,40);(1,2,5,4)(3,6,8,7)(9,11,14,15)(10,12,17,18)(13,19,21,22)(16,20,23,24)(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,40)(37,39);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,18)(16,22)(19,23)(21,24)(25,28,30,38)(26,32,34,36)(27,35,29,37)(31,39,33,40);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7552499);