Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
6675 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 44 points:
(1,2)(3,4)(6,9)(10,12)(13,14)(15,20)(16,21)(17,29)(18,19)(22,30)(23,31)(24,32)(25,39)(26,28)(27,40)(33,36)(34,41)(35,37)(38,44)(42,43);(5,6,8,9)(7,10,11,12)(14,19)(15,23)(21,26)(22,29)(24,35)(25,36)(34,40)(38,43);(2,4)(13,14)(15,20)(16,26)(17,22)(18,19)(21,28)(23,31)(24,37)(25,33)(27,40)(29,30)(32,35)(34,41)(36,39)(38,44)(42,43);(1,2)(3,4)(13,15,18,23)(14,20,19,31)(16,24,28,35)(17,25,30,36)(21,32,26,37)(22,33,29,39)(27,38,41,43)(34,42,40,44);(13,16)(14,21)(15,24)(17,27)(18,28)(19,26)(20,32)(22,34)(23,35)(25,38)(29,40)(30,41)(31,37)(33,42)(36,43)(39,44);(5,7,8,11)(6,10,9,12)(13,17,18,30)(14,22,19,29)(15,25,23,36)(16,27,28,41)(20,33,31,39)(21,34,26,40)(24,38,35,43)(32,42,37,44)Polycyclic presentation:
2^9|b^2=gi,d^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7580740);