Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,8)(10,11)(14,18)(19,20);(1,2)(3,4)(9,10)(11,12)(13,14)(15,19)(16,20)(17,18);(2,4)(9,10,12,11)(13,15,17,16)(14,19,18,20);(6,8)(9,11,12,10)(13,15,17,16)(14,20,18,19);(13,16,17,15)(14,20,18,19)Polycyclic presentation:
2^9|c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72281293, #72281388, #72281393, … |
| maximal normal subgroup of | #72281293, #72281388, #72281393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 516227);