Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(5,6,7,8)(10,13)(14,16)(18,23)(20,25)(24,29)(28,31);(2,4)(11,15)(14,16)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31);(1,2,3,4)(9,10)(11,14)(12,13)(15,16)(17,19,22,27)(18,20,23,25)(21,26,30,32)(24,28,29,31);(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31);(9,10)(11,14)(12,13)(15,16)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72281350, #72281360, #72281371, … |
| maximal normal subgroup of | #72281350, #72281360, #72281371, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 467708);