Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,16,18)(15,19,20,17)(23,27)(26,28);(1,2,3,4)(7,12)(9,11)(13,14)(15,19)(16,18)(17,20);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14,16,18)(15,17,20,19)(21,22,25,24)(23,26,27,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,18)(17,20)(21,23)(22,26)(24,28)(25,27);(5,6)(7,9)(8,10)(11,12)(13,15)(14,17)(16,20)(18,19)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=gi,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3144412);