Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(5,6,8,7)(9,10,14,16,15,20,13,19)(11,17,24,29,22,32,23,33)(12,25,27,34,28,18,26,30)(21,35,38,40,37,31,36,39);(5,7,8,6)(10,16)(12,26)(13,14)(17,29)(19,20)(21,36)(23,24)(27,28)(30,34)(32,33)(37,38)(39,40);(1,2,3,4)(9,11)(10,17)(12,21)(13,23)(14,24)(15,22)(16,29)(18,31)(19,33)(20,32)(25,35)(26,36)(27,38)(28,37)(30,39)(34,40);(9,12,13,26,15,28,14,27)(10,18,19,34,20,25,16,30)(11,21,23,36,22,37,24,38)(17,31,33,40,32,35,29,39);(11,22)(17,32)(21,37)(23,24)(29,33)(31,35)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,f^b=fi,h^b=hi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72343350, #72343358, #72343372, … |
| maximal normal subgroup of | #72343350, #72343358, #72343372, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289980);