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 | #46387 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,13,14)(11,15,17,18)(12,16,20,21)(19,22,23,24)(25,26)(27,33)(28,32)(29,35)(30,31)(34,40)(36,38)(37,39);(2,6)(3,7)(10,14)(11,17)(16,21)(19,23)(25,27)(26,29)(28,36)(30,35)(31,33)(32,40)(34,37)(38,39);(1,3)(2,4)(5,7)(6,8)(11,17)(12,20)(15,18)(16,21)(26,31)(28,37)(29,33)(34,36);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,30,37)(26,32,31,39)(27,34,35,36)(29,38,33,40);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27)(26,29)(28,36)(30,35)(31,33)(32,40)(34,37)(38,39);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,27)(28,38)(30,33)(31,35)(32,34)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72284437, #72284481, #72285088 |
| maximal normal subgroup of | #72284437, #72284481, #72285088 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840309);